Pre-Algebra Examples

$\frac{1}{x + 3} + \frac{3}{y + 7} = \frac{5}{y^{2} + 9 y + 14}$

Step 1

Move all terms not containing $y$ to the right side of the equation.

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Step 1.1

Subtract $\frac{1}{x + 3}$ from both sides of the equation.

$\frac{3}{y + 7} = \frac{5}{y^{2} + 9 y + 14} - \frac{1}{x + 3}$

Step 1.2

Factor $y^{2} + 9 y + 14$ using the AC method.

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Step 1.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $14$ and whose sum is $9$ .

$2, 7$

Step 1.2.2

Write the factored form using these integers.

$\frac{3}{y + 7} = \frac{5}{(y + 2) (y + 7)} - \frac{1}{x + 3}$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$y + 7, (y + 2) (y + 7), x + 3$

Step 2.2

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 2.3

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 2.4

The LCM of $1, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

Step 2.5

The factor for $y + 7$ is $y + 7$ itself.

$(y + 7) = y + 7$

$(y + 7)$ occurs $1$ time.

Step 2.6

The factor for $y + 2$ is $y + 2$ itself.

$(y + 2) = y + 2$

$(y + 2)$ occurs $1$ time.

Step 2.7

The factor for $y + 7$ is $y + 7$ itself.

$(y + 7) = y + 7$

$(y + 7)$ occurs $1$ time.

Step 2.8

The factor for $x + 3$ is $x + 3$ itself.

$(x + 3) = x + 3$

$(x + 3)$ occurs $1$ time.

Step 2.9

The LCM of $y + 7, y + 2, y + 7, x + 3$ is the result of multiplying all factors the greatest number of times they occur in either term.

$(y + 7) (y + 2) (x + 3)$

Step 3

Multiply each term in $\frac{3}{y + 7} = \frac{5}{(y + 2) (y + 7)} - \frac{1}{x + 3}$ by $(y + 7) (y + 2) (x + 3)$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{3}{y + 7} = \frac{5}{(y + 2) (y + 7)} - \frac{1}{x + 3}$ by $(y + 7) (y + 2) (x + 3)$ .

$\frac{3}{y + 7} ((y + 7) (y + 2) (x + 3)) = \frac{5}{(y + 2) (y + 7)} ((y + 7) (y + 2) (x + 3)) - \frac{1}{x + 3} ((y + 7) (y + 2) (x + 3))$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $y + 7$ .

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Step 3.2.1.1

Factor $y + 7$ out of $(y + 7) (y + 2) (x + 3)$ .

$\frac{3}{y + 7} ((y + 7) ((y + 2) (x + 3))) = \frac{5}{(y + 2) (y + 7)} ((y + 7) (y + 2) (x + 3)) - \frac{1}{x + 3} ((y + 7) (y + 2) (x + 3))$

Step 3.2.1.2

Cancel the common factor.

$\frac{3}{y + 7} ((y + 7) ((y + 2) (x + 3))) = \frac{5}{(y + 2) (y + 7)} ((y + 7) (y + 2) (x + 3)) - \frac{1}{x + 3} ((y + 7) (y + 2) (x + 3))$

Step 3.2.1.3

Rewrite the expression.

$3 ((y + 2) (x + 3)) = \frac{5}{(y + 2) (y + 7)} ((y + 7) (y + 2) (x + 3)) - \frac{1}{x + 3} ((y + 7) (y + 2) (x + 3))$

Step 3.2.2

Expand $(y + 2) (x + 3)$ using the FOIL Method.

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Step 3.2.2.1

Apply the distributive property.

$3 (y (x + 3) + 2 (x + 3)) = \frac{5}{(y + 2) (y + 7)} ((y + 7) (y + 2) (x + 3)) - \frac{1}{x + 3} ((y + 7) (y + 2) (x + 3))$

Step 3.2.2.2

Apply the distributive property.

$3 (y x + y \cdot 3 + 2 (x + 3)) = \frac{5}{(y + 2) (y + 7)} ((y + 7) (y + 2) (x + 3)) - \frac{1}{x + 3} ((y + 7) (y + 2) (x + 3))$

Step 3.2.2.3

Apply the distributive property.

$3 (y x + y \cdot 3 + 2 x + 2 \cdot 3) = \frac{5}{(y + 2) (y + 7)} ((y + 7) (y + 2) (x + 3)) - \frac{1}{x + 3} ((y + 7) (y + 2) (x + 3))$

Step 3.2.3

Simplify terms.

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Move $3$ to the left of $y$ .

$3 (y x + 3 \cdot y + 2 x + 2 \cdot 3) = \frac{5}{(y + 2) (y + 7)} ((y + 7) (y + 2) (x + 3)) - \frac{1}{x + 3} ((y + 7) (y + 2) (x + 3))$

Step 3.2.3.1.2

Multiply $2$ by $3$ .

$3 (y x + 3 y + 2 x + 6) = \frac{5}{(y + 2) (y + 7)} ((y + 7) (y + 2) (x + 3)) - \frac{1}{x + 3} ((y + 7) (y + 2) (x + 3))$

Step 3.2.3.2

Apply the distributive property.

$3 (y x) + 3 (3 y) + 3 (2 x) + 3 \cdot 6 = \frac{5}{(y + 2) (y + 7)} ((y + 7) (y + 2) (x + 3)) - \frac{1}{x + 3} ((y + 7) (y + 2) (x + 3))$

Step 3.2.4

Simplify.

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Step 3.2.4.1

Multiply $3$ by $3$ .

$3 y x + 9 y + 3 (2 x) + 3 \cdot 6 = \frac{5}{(y + 2) (y + 7)} ((y + 7) (y + 2) (x + 3)) - \frac{1}{x + 3} ((y + 7) (y + 2) (x + 3))$

Step 3.2.4.2

Multiply $2$ by $3$ .

$3 y x + 9 y + 6 x + 3 \cdot 6 = \frac{5}{(y + 2) (y + 7)} ((y + 7) (y + 2) (x + 3)) - \frac{1}{x + 3} ((y + 7) (y + 2) (x + 3))$

Step 3.2.4.3

Multiply $3$ by $6$ .

$3 y x + 9 y + 6 x + 18 = \frac{5}{(y + 2) (y + 7)} ((y + 7) (y + 2) (x + 3)) - \frac{1}{x + 3} ((y + 7) (y + 2) (x + 3))$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Cancel the common factor of $(y + 7) (y + 2)$ .

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Step 3.3.1.1.1

Factor $(y + 7) (y + 2)$ out of $(y + 2) (y + 7)$ .

$3 y x + 9 y + 6 x + 18 = \frac{5}{(y + 7) (y + 2) (1)} ((y + 7) (y + 2) (x + 3)) - \frac{1}{x + 3} ((y + 7) (y + 2) (x + 3))$

Step 3.3.1.1.2

Cancel the common factor.

$3 y x + 9 y + 6 x + 18 = \frac{5}{(y + 7) (y + 2) \cdot 1} ((y + 7) (y + 2) (x + 3)) - \frac{1}{x + 3} ((y + 7) (y + 2) (x + 3))$

Step 3.3.1.1.3

Rewrite the expression.

$3 y x + 9 y + 6 x + 18 = 5 (x + 3) - \frac{1}{x + 3} ((y + 7) (y + 2) (x + 3))$

Step 3.3.1.2

Apply the distributive property.

$3 y x + 9 y + 6 x + 18 = 5 x + 5 \cdot 3 - \frac{1}{x + 3} ((y + 7) (y + 2) (x + 3))$

Step 3.3.1.3

Multiply $5$ by $3$ .

$3 y x + 9 y + 6 x + 18 = 5 x + 15 - \frac{1}{x + 3} ((y + 7) (y + 2) (x + 3))$

Step 3.3.1.4

Cancel the common factor of $x + 3$ .

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Step 3.3.1.4.1

Move the leading negative in $- \frac{1}{x + 3}$ into the numerator.

$3 y x + 9 y + 6 x + 18 = 5 x + 15 + \frac{- 1}{x + 3} ((y + 7) (y + 2) (x + 3))$

Step 3.3.1.4.2

Factor $x + 3$ out of $(y + 7) (y + 2) (x + 3)$ .

$3 y x + 9 y + 6 x + 18 = 5 x + 15 + \frac{- 1}{x + 3} ((x + 3) ((y + 7) (y + 2)))$

Step 3.3.1.4.3

Cancel the common factor.

$3 y x + 9 y + 6 x + 18 = 5 x + 15 + \frac{- 1}{x + 3} ((x + 3) ((y + 7) (y + 2)))$

Step 3.3.1.4.4

Rewrite the expression.

$3 y x + 9 y + 6 x + 18 = 5 x + 15 - ((y + 7) (y + 2))$

Step 3.3.1.5

Expand $(y + 7) (y + 2)$ using the FOIL Method.

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Step 3.3.1.5.1

Apply the distributive property.

$3 y x + 9 y + 6 x + 18 = 5 x + 15 - (y (y + 2) + 7 (y + 2))$

Step 3.3.1.5.2

Apply the distributive property.

$3 y x + 9 y + 6 x + 18 = 5 x + 15 - (y \cdot y + y \cdot 2 + 7 (y + 2))$

Step 3.3.1.5.3

Apply the distributive property.

$3 y x + 9 y + 6 x + 18 = 5 x + 15 - (y \cdot y + y \cdot 2 + 7 y + 7 \cdot 2)$

Step 3.3.1.6

Simplify and combine like terms.

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Step 3.3.1.6.1

Simplify each term.

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Step 3.3.1.6.1.1

Multiply $y$ by $y$ .

$3 y x + 9 y + 6 x + 18 = 5 x + 15 - (y^{2} + y \cdot 2 + 7 y + 7 \cdot 2)$

Step 3.3.1.6.1.2

Move $2$ to the left of $y$ .

$3 y x + 9 y + 6 x + 18 = 5 x + 15 - (y^{2} + 2 \cdot y + 7 y + 7 \cdot 2)$

Step 3.3.1.6.1.3

Multiply $7$ by $2$ .

$3 y x + 9 y + 6 x + 18 = 5 x + 15 - (y^{2} + 2 y + 7 y + 14)$

Step 3.3.1.6.2

Add $2 y$ and $7 y$ .

$3 y x + 9 y + 6 x + 18 = 5 x + 15 - (y^{2} + 9 y + 14)$

Step 3.3.1.7

Apply the distributive property.

$3 y x + 9 y + 6 x + 18 = 5 x + 15 - y^{2} - (9 y) - 1 \cdot 14$

Step 3.3.1.8

Simplify.

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Step 3.3.1.8.1

Multiply $9$ by $- 1$ .

$3 y x + 9 y + 6 x + 18 = 5 x + 15 - y^{2} - 9 y - 1 \cdot 14$

Step 3.3.1.8.2

Multiply $- 1$ by $14$ .

$3 y x + 9 y + 6 x + 18 = 5 x + 15 - y^{2} - 9 y - 14$

Step 3.3.2

Subtract $14$ from $15$ .

$3 y x + 9 y + 6 x + 18 = 5 x - y^{2} - 9 y + 1$

Step 4

Solve the equation.

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Step 4.1

Move all terms containing $y$ to the left side of the equation.

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Step 4.1.1

Add $y^{2}$ to both sides of the equation.

$3 y x + 9 y + 6 x + 18 + y^{2} = 5 x - 9 y + 1$

Step 4.1.2

Add $9 y$ to both sides of the equation.

$3 y x + 9 y + 6 x + 18 + y^{2} + 9 y = 5 x + 1$

Step 4.1.3

Add $9 y$ and $9 y$ .

$3 y x + 6 x + 18 + y^{2} + 18 y = 5 x + 1$

Step 4.2

Move all terms to the left side of the equation and simplify.

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Step 4.2.1

Move all the expressions to the left side of the equation.

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Step 4.2.1.1

Subtract $5 x$ from both sides of the equation.

$3 y x + 6 x + 18 + y^{2} + 18 y - 5 x = 1$

Step 4.2.1.2

Subtract $1$ from both sides of the equation.

$3 y x + 6 x + 18 + y^{2} + 18 y - 5 x - 1 = 0$

Step 4.2.2

Simplify $3 y x + 6 x + 18 + y^{2} + 18 y - 5 x - 1$ .

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Step 4.2.2.1

Subtract $5 x$ from $6 x$ .

$3 y x + x + 18 + y^{2} + 18 y - 1 = 0$

Step 4.2.2.2

Subtract $1$ from $18$ .

$3 y x + x + y^{2} + 18 y + 17 = 0$

Step 4.3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4.4

Substitute the values $a = 1$ , $b = 3 x + 18$ , and $c = x + 17$ into the quadratic formula and solve for $y$ .

$\frac{- (3 x + 18) \pm \sqrt{{(3 x + 18)}^{2} - 4 \cdot (1 \cdot (x + 17))}}{2 \cdot 1}$

Step 4.5

Simplify.

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Step 4.5.1

Simplify the numerator.

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Step 4.5.1.1

Apply the distributive property.

$y = \frac{- (3 x) - 1 \cdot 18 \pm \sqrt{{(3 x + 18)}^{2} - 4 \cdot 1 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.5.1.2

Multiply $3$ by $- 1$ .

$y = \frac{- 3 x - 1 \cdot 18 \pm \sqrt{{(3 x + 18)}^{2} - 4 \cdot 1 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.5.1.3

Multiply $- 1$ by $18$ .

$y = \frac{- 3 x - 18 \pm \sqrt{{(3 x + 18)}^{2} - 4 \cdot 1 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.5.1.4

Rewrite ${(3 x + 18)}^{2}$ as $(3 x + 18) (3 x + 18)$ .

$y = \frac{- 3 x - 18 \pm \sqrt{(3 x + 18) (3 x + 18) - 4 \cdot 1 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.5.1.5

Expand $(3 x + 18) (3 x + 18)$ using the FOIL Method.

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Step 4.5.1.5.1

Apply the distributive property.

$y = \frac{- 3 x - 18 \pm \sqrt{3 x (3 x + 18) + 18 (3 x + 18) - 4 \cdot 1 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.5.1.5.2

Apply the distributive property.

$y = \frac{- 3 x - 18 \pm \sqrt{3 x (3 x) + 3 x \cdot 18 + 18 (3 x + 18) - 4 \cdot 1 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.5.1.5.3

Apply the distributive property.

$y = \frac{- 3 x - 18 \pm \sqrt{3 x (3 x) + 3 x \cdot 18 + 18 (3 x) + 18 \cdot 18 - 4 \cdot 1 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.5.1.6

Simplify and combine like terms.

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Step 4.5.1.6.1

Simplify each term.

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Step 4.5.1.6.1.1

Rewrite using the commutative property of multiplication.

$y = \frac{- 3 x - 18 \pm \sqrt{3 \cdot (3 x \cdot x) + 3 x \cdot 18 + 18 (3 x) + 18 \cdot 18 - 4 \cdot 1 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.5.1.6.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 4.5.1.6.1.2.1

Move $x$ .

$y = \frac{- 3 x - 18 \pm \sqrt{3 \cdot (3 (x \cdot x)) + 3 x \cdot 18 + 18 (3 x) + 18 \cdot 18 - 4 \cdot 1 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.5.1.6.1.2.2

Multiply $x$ by $x$ .

$y = \frac{- 3 x - 18 \pm \sqrt{3 \cdot (3 x^{2}) + 3 x \cdot 18 + 18 (3 x) + 18 \cdot 18 - 4 \cdot 1 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.5.1.6.1.3

Multiply $3$ by $3$ .

$y = \frac{- 3 x - 18 \pm \sqrt{9 x^{2} + 3 x \cdot 18 + 18 (3 x) + 18 \cdot 18 - 4 \cdot 1 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.5.1.6.1.4

Multiply $18$ by $3$ .

$y = \frac{- 3 x - 18 \pm \sqrt{9 x^{2} + 54 x + 18 (3 x) + 18 \cdot 18 - 4 \cdot 1 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.5.1.6.1.5

Multiply $3$ by $18$ .

$y = \frac{- 3 x - 18 \pm \sqrt{9 x^{2} + 54 x + 54 x + 18 \cdot 18 - 4 \cdot 1 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.5.1.6.1.6

Multiply $18$ by $18$ .

$y = \frac{- 3 x - 18 \pm \sqrt{9 x^{2} + 54 x + 54 x + 324 - 4 \cdot 1 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.5.1.6.2

Add $54 x$ and $54 x$ .

$y = \frac{- 3 x - 18 \pm \sqrt{9 x^{2} + 108 x + 324 - 4 \cdot 1 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.5.1.7

Multiply $- 4$ by $1$ .

$y = \frac{- 3 x - 18 \pm \sqrt{9 x^{2} + 108 x + 324 - 4 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.5.1.8

Apply the distributive property.

$y = \frac{- 3 x - 18 \pm \sqrt{9 x^{2} + 108 x + 324 - 4 x - 4 \cdot 17}}{2 \cdot 1}$

Step 4.5.1.9

Multiply $- 4$ by $17$ .

$y = \frac{- 3 x - 18 \pm \sqrt{9 x^{2} + 108 x + 324 - 4 x - 68}}{2 \cdot 1}$

Step 4.5.1.10

Subtract $4 x$ from $108 x$ .

$y = \frac{- 3 x - 18 \pm \sqrt{9 x^{2} + 104 x + 324 - 68}}{2 \cdot 1}$

Step 4.5.1.11

Subtract $68$ from $324$ .

$y = \frac{- 3 x - 18 \pm \sqrt{9 x^{2} + 104 x + 256}}{2 \cdot 1}$

Step 4.5.1.12

Factor by grouping.

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Step 4.5.1.12.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 9 \cdot 256 = 2304$ and whose sum is $b = 104$ .

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Step 4.5.1.12.1.1

Factor $104$ out of $104 x$ .

$y = \frac{- 3 x - 18 \pm \sqrt{9 x^{2} + 104 (x) + 256}}{2 \cdot 1}$

Step 4.5.1.12.1.2

Rewrite $104$ as $32$ plus $72$

$y = \frac{- 3 x - 18 \pm \sqrt{9 x^{2} + (32 + 72) x + 256}}{2 \cdot 1}$

Step 4.5.1.12.1.3

Apply the distributive property.

$y = \frac{- 3 x - 18 \pm \sqrt{9 x^{2} + 32 x + 72 x + 256}}{2 \cdot 1}$

Step 4.5.1.12.2

Factor out the greatest common factor from each group.

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Step 4.5.1.12.2.1

Group the first two terms and the last two terms.

$y = \frac{- 3 x - 18 \pm \sqrt{(9 x^{2} + 32 x) + 72 x + 256}}{2 \cdot 1}$

Step 4.5.1.12.2.2

Factor out the greatest common factor (GCF) from each group.

$y = \frac{- 3 x - 18 \pm \sqrt{x (9 x + 32) + 8 (9 x + 32)}}{2 \cdot 1}$

Step 4.5.1.12.3

Factor the polynomial by factoring out the greatest common factor, $9 x + 32$ .

$y = \frac{- 3 x - 18 \pm \sqrt{(9 x + 32) (x + 8)}}{2 \cdot 1}$

Step 4.5.2

Multiply $2$ by $1$ .

$y = \frac{- 3 x - 18 \pm \sqrt{(9 x + 32) (x + 8)}}{2}$

Step 4.6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 4.6.1

Simplify the numerator.

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Step 4.6.1.1

Apply the distributive property.

$y = \frac{- (3 x) - 1 \cdot 18 \pm \sqrt{{(3 x + 18)}^{2} - 4 \cdot 1 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.6.1.2

Multiply $3$ by $- 1$ .

$y = \frac{- 3 x - 1 \cdot 18 \pm \sqrt{{(3 x + 18)}^{2} - 4 \cdot 1 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.6.1.3

Multiply $- 1$ by $18$ .

$y = \frac{- 3 x - 18 \pm \sqrt{{(3 x + 18)}^{2} - 4 \cdot 1 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.6.1.4

Rewrite ${(3 x + 18)}^{2}$ as $(3 x + 18) (3 x + 18)$ .

$y = \frac{- 3 x - 18 \pm \sqrt{(3 x + 18) (3 x + 18) - 4 \cdot 1 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.6.1.5

Expand $(3 x + 18) (3 x + 18)$ using the FOIL Method.

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Step 4.6.1.5.1

Apply the distributive property.

$y = \frac{- 3 x - 18 \pm \sqrt{3 x (3 x + 18) + 18 (3 x + 18) - 4 \cdot 1 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.6.1.5.2

Apply the distributive property.

$y = \frac{- 3 x - 18 \pm \sqrt{3 x (3 x) + 3 x \cdot 18 + 18 (3 x + 18) - 4 \cdot 1 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.6.1.5.3

Apply the distributive property.

$y = \frac{- 3 x - 18 \pm \sqrt{3 x (3 x) + 3 x \cdot 18 + 18 (3 x) + 18 \cdot 18 - 4 \cdot 1 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.6.1.6

Simplify and combine like terms.

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Step 4.6.1.6.1

Simplify each term.

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Step 4.6.1.6.1.1

Rewrite using the commutative property of multiplication.

$y = \frac{- 3 x - 18 \pm \sqrt{3 \cdot (3 x \cdot x) + 3 x \cdot 18 + 18 (3 x) + 18 \cdot 18 - 4 \cdot 1 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.6.1.6.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 4.6.1.6.1.2.1

Move $x$ .

$y = \frac{- 3 x - 18 \pm \sqrt{3 \cdot (3 (x \cdot x)) + 3 x \cdot 18 + 18 (3 x) + 18 \cdot 18 - 4 \cdot 1 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.6.1.6.1.2.2

Multiply $x$ by $x$ .

$y = \frac{- 3 x - 18 \pm \sqrt{3 \cdot (3 x^{2}) + 3 x \cdot 18 + 18 (3 x) + 18 \cdot 18 - 4 \cdot 1 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.6.1.6.1.3

Multiply $3$ by $3$ .

$y = \frac{- 3 x - 18 \pm \sqrt{9 x^{2} + 3 x \cdot 18 + 18 (3 x) + 18 \cdot 18 - 4 \cdot 1 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.6.1.6.1.4

Multiply $18$ by $3$ .

$y = \frac{- 3 x - 18 \pm \sqrt{9 x^{2} + 54 x + 18 (3 x) + 18 \cdot 18 - 4 \cdot 1 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.6.1.6.1.5

Multiply $3$ by $18$ .

$y = \frac{- 3 x - 18 \pm \sqrt{9 x^{2} + 54 x + 54 x + 18 \cdot 18 - 4 \cdot 1 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.6.1.6.1.6

Multiply $18$ by $18$ .

$y = \frac{- 3 x - 18 \pm \sqrt{9 x^{2} + 54 x + 54 x + 324 - 4 \cdot 1 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.6.1.6.2

Add $54 x$ and $54 x$ .

$y = \frac{- 3 x - 18 \pm \sqrt{9 x^{2} + 108 x + 324 - 4 \cdot 1 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.6.1.7

Multiply $- 4$ by $1$ .

$y = \frac{- 3 x - 18 \pm \sqrt{9 x^{2} + 108 x + 324 - 4 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.6.1.8

Apply the distributive property.

$y = \frac{- 3 x - 18 \pm \sqrt{9 x^{2} + 108 x + 324 - 4 x - 4 \cdot 17}}{2 \cdot 1}$

Step 4.6.1.9

Multiply $- 4$ by $17$ .

$y = \frac{- 3 x - 18 \pm \sqrt{9 x^{2} + 108 x + 324 - 4 x - 68}}{2 \cdot 1}$

Step 4.6.1.10

Subtract $4 x$ from $108 x$ .

$y = \frac{- 3 x - 18 \pm \sqrt{9 x^{2} + 104 x + 324 - 68}}{2 \cdot 1}$

Step 4.6.1.11

Subtract $68$ from $324$ .

$y = \frac{- 3 x - 18 \pm \sqrt{9 x^{2} + 104 x + 256}}{2 \cdot 1}$

Step 4.6.1.12

Factor by grouping.

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Step 4.6.1.12.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 9 \cdot 256 = 2304$ and whose sum is $b = 104$ .

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Step 4.6.1.12.1.1

Factor $104$ out of $104 x$ .

$y = \frac{- 3 x - 18 \pm \sqrt{9 x^{2} + 104 (x) + 256}}{2 \cdot 1}$

Step 4.6.1.12.1.2

Rewrite $104$ as $32$ plus $72$

$y = \frac{- 3 x - 18 \pm \sqrt{9 x^{2} + (32 + 72) x + 256}}{2 \cdot 1}$

Step 4.6.1.12.1.3

Apply the distributive property.

$y = \frac{- 3 x - 18 \pm \sqrt{9 x^{2} + 32 x + 72 x + 256}}{2 \cdot 1}$

Step 4.6.1.12.2

Factor out the greatest common factor from each group.

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Step 4.6.1.12.2.1

Group the first two terms and the last two terms.

$y = \frac{- 3 x - 18 \pm \sqrt{(9 x^{2} + 32 x) + 72 x + 256}}{2 \cdot 1}$

Step 4.6.1.12.2.2

Factor out the greatest common factor (GCF) from each group.

$y = \frac{- 3 x - 18 \pm \sqrt{x (9 x + 32) + 8 (9 x + 32)}}{2 \cdot 1}$

Step 4.6.1.12.3

Factor the polynomial by factoring out the greatest common factor, $9 x + 32$ .

$y = \frac{- 3 x - 18 \pm \sqrt{(9 x + 32) (x + 8)}}{2 \cdot 1}$

Step 4.6.2

Multiply $2$ by $1$ .

$y = \frac{- 3 x - 18 \pm \sqrt{(9 x + 32) (x + 8)}}{2}$

Step 4.6.3

Change the $\pm$ to $+$ .

$y = \frac{- 3 x - 18 + \sqrt{(9 x + 32) (x + 8)}}{2}$

Step 4.6.4

Factor $- 1$ out of $- 3 x$ .

$y = \frac{- (3 x) - 18 + \sqrt{(9 x + 32) (x + 8)}}{2}$

Step 4.6.5

Rewrite $- 18$ as $- 1 (18)$ .

$y = \frac{- (3 x) - 1 \cdot 18 + \sqrt{(9 x + 32) (x + 8)}}{2}$

Step 4.6.6

Factor $- 1$ out of $- (3 x) - 1 (18)$ .

$y = \frac{- (3 x + 18) + \sqrt{(9 x + 32) (x + 8)}}{2}$

Step 4.6.7

Factor $- 1$ out of $\sqrt{(9 x + 32) (x + 8)}$ .

$y = \frac{- (3 x + 18) - 1 (- \sqrt{(9 x + 32) (x + 8)})}{2}$

Step 4.6.8

Factor $- 1$ out of $- (3 x + 18) - 1 (- \sqrt{(9 x + 32) (x + 8)})$ .

$y = \frac{- (3 x + 18 - \sqrt{(9 x + 32) (x + 8)})}{2}$

Step 4.6.9

Rewrite $- (3 x + 18 - \sqrt{(9 x + 32) (x + 8)})$ as $- 1 (3 x + 18 - \sqrt{(9 x + 32) (x + 8)})$ .

$y = \frac{- 1 (3 x + 18 - \sqrt{(9 x + 32) (x + 8)})}{2}$

Step 4.6.10

Move the negative in front of the fraction.

$y = - \frac{3 x + 18 - \sqrt{(9 x + 32) (x + 8)}}{2}$

Step 4.7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 4.7.1

Simplify the numerator.

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Step 4.7.1.1

Apply the distributive property.

$y = \frac{- (3 x) - 1 \cdot 18 \pm \sqrt{{(3 x + 18)}^{2} - 4 \cdot 1 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.7.1.2

Multiply $3$ by $- 1$ .

$y = \frac{- 3 x - 1 \cdot 18 \pm \sqrt{{(3 x + 18)}^{2} - 4 \cdot 1 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.7.1.3

Multiply $- 1$ by $18$ .

$y = \frac{- 3 x - 18 \pm \sqrt{{(3 x + 18)}^{2} - 4 \cdot 1 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.7.1.4

Rewrite ${(3 x + 18)}^{2}$ as $(3 x + 18) (3 x + 18)$ .

$y = \frac{- 3 x - 18 \pm \sqrt{(3 x + 18) (3 x + 18) - 4 \cdot 1 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.7.1.5

Expand $(3 x + 18) (3 x + 18)$ using the FOIL Method.

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Step 4.7.1.5.1

Apply the distributive property.

$y = \frac{- 3 x - 18 \pm \sqrt{3 x (3 x + 18) + 18 (3 x + 18) - 4 \cdot 1 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.7.1.5.2

Apply the distributive property.

$y = \frac{- 3 x - 18 \pm \sqrt{3 x (3 x) + 3 x \cdot 18 + 18 (3 x + 18) - 4 \cdot 1 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.7.1.5.3

Apply the distributive property.

$y = \frac{- 3 x - 18 \pm \sqrt{3 x (3 x) + 3 x \cdot 18 + 18 (3 x) + 18 \cdot 18 - 4 \cdot 1 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.7.1.6

Simplify and combine like terms.

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Step 4.7.1.6.1

Simplify each term.

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Step 4.7.1.6.1.1

Rewrite using the commutative property of multiplication.

$y = \frac{- 3 x - 18 \pm \sqrt{3 \cdot (3 x \cdot x) + 3 x \cdot 18 + 18 (3 x) + 18 \cdot 18 - 4 \cdot 1 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.7.1.6.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 4.7.1.6.1.2.1

Move $x$ .

$y = \frac{- 3 x - 18 \pm \sqrt{3 \cdot (3 (x \cdot x)) + 3 x \cdot 18 + 18 (3 x) + 18 \cdot 18 - 4 \cdot 1 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.7.1.6.1.2.2

Multiply $x$ by $x$ .

$y = \frac{- 3 x - 18 \pm \sqrt{3 \cdot (3 x^{2}) + 3 x \cdot 18 + 18 (3 x) + 18 \cdot 18 - 4 \cdot 1 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.7.1.6.1.3

Multiply $3$ by $3$ .

$y = \frac{- 3 x - 18 \pm \sqrt{9 x^{2} + 3 x \cdot 18 + 18 (3 x) + 18 \cdot 18 - 4 \cdot 1 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.7.1.6.1.4

Multiply $18$ by $3$ .

$y = \frac{- 3 x - 18 \pm \sqrt{9 x^{2} + 54 x + 18 (3 x) + 18 \cdot 18 - 4 \cdot 1 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.7.1.6.1.5

Multiply $3$ by $18$ .

$y = \frac{- 3 x - 18 \pm \sqrt{9 x^{2} + 54 x + 54 x + 18 \cdot 18 - 4 \cdot 1 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.7.1.6.1.6

Multiply $18$ by $18$ .

$y = \frac{- 3 x - 18 \pm \sqrt{9 x^{2} + 54 x + 54 x + 324 - 4 \cdot 1 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.7.1.6.2

Add $54 x$ and $54 x$ .

$y = \frac{- 3 x - 18 \pm \sqrt{9 x^{2} + 108 x + 324 - 4 \cdot 1 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.7.1.7

Multiply $- 4$ by $1$ .

$y = \frac{- 3 x - 18 \pm \sqrt{9 x^{2} + 108 x + 324 - 4 \cdot (x + 17)}}{2 \cdot 1}$

Step 4.7.1.8

Apply the distributive property.

$y = \frac{- 3 x - 18 \pm \sqrt{9 x^{2} + 108 x + 324 - 4 x - 4 \cdot 17}}{2 \cdot 1}$

Step 4.7.1.9

Multiply $- 4$ by $17$ .

$y = \frac{- 3 x - 18 \pm \sqrt{9 x^{2} + 108 x + 324 - 4 x - 68}}{2 \cdot 1}$

Step 4.7.1.10

Subtract $4 x$ from $108 x$ .

$y = \frac{- 3 x - 18 \pm \sqrt{9 x^{2} + 104 x + 324 - 68}}{2 \cdot 1}$

Step 4.7.1.11

Subtract $68$ from $324$ .

$y = \frac{- 3 x - 18 \pm \sqrt{9 x^{2} + 104 x + 256}}{2 \cdot 1}$

Step 4.7.1.12

Factor by grouping.

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Step 4.7.1.12.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 9 \cdot 256 = 2304$ and whose sum is $b = 104$ .

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Step 4.7.1.12.1.1

Factor $104$ out of $104 x$ .

$y = \frac{- 3 x - 18 \pm \sqrt{9 x^{2} + 104 (x) + 256}}{2 \cdot 1}$

Step 4.7.1.12.1.2

Rewrite $104$ as $32$ plus $72$

$y = \frac{- 3 x - 18 \pm \sqrt{9 x^{2} + (32 + 72) x + 256}}{2 \cdot 1}$

Step 4.7.1.12.1.3

Apply the distributive property.

$y = \frac{- 3 x - 18 \pm \sqrt{9 x^{2} + 32 x + 72 x + 256}}{2 \cdot 1}$

Step 4.7.1.12.2

Factor out the greatest common factor from each group.

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Step 4.7.1.12.2.1

Group the first two terms and the last two terms.

$y = \frac{- 3 x - 18 \pm \sqrt{(9 x^{2} + 32 x) + 72 x + 256}}{2 \cdot 1}$

Step 4.7.1.12.2.2

Factor out the greatest common factor (GCF) from each group.

$y = \frac{- 3 x - 18 \pm \sqrt{x (9 x + 32) + 8 (9 x + 32)}}{2 \cdot 1}$

Step 4.7.1.12.3

Factor the polynomial by factoring out the greatest common factor, $9 x + 32$ .

$y = \frac{- 3 x - 18 \pm \sqrt{(9 x + 32) (x + 8)}}{2 \cdot 1}$

Step 4.7.2

Multiply $2$ by $1$ .

$y = \frac{- 3 x - 18 \pm \sqrt{(9 x + 32) (x + 8)}}{2}$

Step 4.7.3

Change the $\pm$ to $-$ .

$y = \frac{- 3 x - 18 - \sqrt{(9 x + 32) (x + 8)}}{2}$

Step 4.7.4

Factor $- 1$ out of $- 3 x$ .

$y = \frac{- (3 x) - 18 - \sqrt{(9 x + 32) (x + 8)}}{2}$

Step 4.7.5

Rewrite $- 18$ as $- 1 (18)$ .

$y = \frac{- (3 x) - 1 \cdot 18 - \sqrt{(9 x + 32) (x + 8)}}{2}$

Step 4.7.6

Factor $- 1$ out of $- (3 x) - 1 (18)$ .

$y = \frac{- (3 x + 18) - \sqrt{(9 x + 32) (x + 8)}}{2}$

Step 4.7.7

Factor $- 1$ out of $- \sqrt{(9 x + 32) (x + 8)}$ .

$y = \frac{- (3 x + 18) - (\sqrt{(9 x + 32) (x + 8)})}{2}$

Step 4.7.8

Factor $- 1$ out of $- (3 x + 18) - (\sqrt{(9 x + 32) (x + 8)})$ .

$y = \frac{- (3 x + 18 + \sqrt{(9 x + 32) (x + 8)})}{2}$

Step 4.7.9

Rewrite $- (3 x + 18 + \sqrt{(9 x + 32) (x + 8)})$ as $- 1 (3 x + 18 + \sqrt{(9 x + 32) (x + 8)})$ .

$y = \frac{- 1 (3 x + 18 + \sqrt{(9 x + 32) (x + 8)})}{2}$

Step 4.7.10

Move the negative in front of the fraction.

$y = - \frac{3 x + 18 + \sqrt{(9 x + 32) (x + 8)}}{2}$

Step 4.8

The final answer is the combination of both solutions.

$y = - \frac{3 x + 18 - \sqrt{(9 x + 32) (x + 8)}}{2}$

$y = - \frac{3 x + 18 + \sqrt{(9 x + 32) (x + 8)}}{2}$

$y = - \frac{3 x + 18 - \sqrt{(9 x + 32) (x + 8)}}{2}$

$y = - \frac{3 x + 18 + \sqrt{(9 x + 32) (x + 8)}}{2}$

Step 5

The given equation $y = - \frac{3 x + 18 - \sqrt{(9 x + 32) (x + 8)}}{2}, - \frac{3 x + 18 + \sqrt{(9 x + 32) (x + 8)}}{2}$ can not be written as $y = k x^{2}$ , so $y$ doesn't vary directly with $x^{2}$ .

$y$ doesn't vary directly with $x$