Precalculus Examples

$14 y + y^{2} = 4 x - 97$

Step 1

Solve for $y$ .

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Step 1.1

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

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Step 1.1.1

Subtract $4 x$ from both sides of the equation.

$14 y + y^{2} - 4 x = - 97$

Step 1.1.2

Add $97$ to both sides of the equation.

$14 y + y^{2} - 4 x + 97 = 0$

Step 1.2

Use the quadratic formula to find the solutions.

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

Step 1.3

Substitute the values $a = 1$ , $b = 14$ , and $c = - 4 x + 97$ into the quadratic formula and solve for $y$ .

$\frac{- 14 \pm \sqrt{14^{2} - 4 \cdot (1 \cdot (- 4 x + 97))}}{2 \cdot 1}$

Step 1.4

Simplify.

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Step 1.4.1

Simplify the numerator.

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Step 1.4.1.1

Raise $14$ to the power of $2$ .

$y = \frac{- 14 \pm \sqrt{196 - 4 \cdot 1 \cdot (- 4 x + 97)}}{2 \cdot 1}$

Step 1.4.1.2

Multiply $- 4$ by $1$ .

$y = \frac{- 14 \pm \sqrt{196 - 4 \cdot (- 4 x + 97)}}{2 \cdot 1}$

Step 1.4.1.3

Apply the distributive property.

$y = \frac{- 14 \pm \sqrt{196 - 4 (- 4 x) - 4 \cdot 97}}{2 \cdot 1}$

Step 1.4.1.4

Multiply $- 4$ by $- 4$ .

$y = \frac{- 14 \pm \sqrt{196 + 16 x - 4 \cdot 97}}{2 \cdot 1}$

Step 1.4.1.5

Multiply $- 4$ by $97$ .

$y = \frac{- 14 \pm \sqrt{196 + 16 x - 388}}{2 \cdot 1}$

Step 1.4.1.6

Subtract $388$ from $196$ .

$y = \frac{- 14 \pm \sqrt{16 x - 192}}{2 \cdot 1}$

Step 1.4.1.7

Factor $16$ out of $16 x - 192$ .

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Step 1.4.1.7.1

Factor $16$ out of $16 x$ .

$y = \frac{- 14 \pm \sqrt{16 (x) - 192}}{2 \cdot 1}$

Step 1.4.1.7.2

Factor $16$ out of $- 192$ .

$y = \frac{- 14 \pm \sqrt{16 x + 16 \cdot - 12}}{2 \cdot 1}$

Step 1.4.1.7.3

Factor $16$ out of $16 x + 16 \cdot - 12$ .

$y = \frac{- 14 \pm \sqrt{16 (x - 12)}}{2 \cdot 1}$

Step 1.4.1.8

Rewrite $16 (x - 12)$ as ${(2^{2})}^{2} (x - 12)$ .

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Step 1.4.1.8.1

Rewrite $16$ as $4^{2}$ .

$y = \frac{- 14 \pm \sqrt{4^{2} (x - 12)}}{2 \cdot 1}$

Step 1.4.1.8.2

Rewrite $4$ as $2^{2}$ .

$y = \frac{- 14 \pm \sqrt{{(2^{2})}^{2} (x - 12)}}{2 \cdot 1}$

Step 1.4.1.9

Pull terms out from under the radical.

$y = \frac{- 14 \pm 2^{2} \sqrt{x - 12}}{2 \cdot 1}$

Step 1.4.1.10

Raise $2$ to the power of $2$ .

$y = \frac{- 14 \pm 4 \sqrt{x - 12}}{2 \cdot 1}$

Step 1.4.2

Multiply $2$ by $1$ .

$y = \frac{- 14 \pm 4 \sqrt{x - 12}}{2}$

Step 1.4.3

Simplify $\frac{- 14 \pm 4 \sqrt{x - 12}}{2}$ .

$y = - 7 \pm 2 \sqrt{x - 12}$

Step 1.5

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

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Step 1.5.1

Simplify the numerator.

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Step 1.5.1.1

Raise $14$ to the power of $2$ .

$y = \frac{- 14 \pm \sqrt{196 - 4 \cdot 1 \cdot (- 4 x + 97)}}{2 \cdot 1}$

Step 1.5.1.2

Multiply $- 4$ by $1$ .

$y = \frac{- 14 \pm \sqrt{196 - 4 \cdot (- 4 x + 97)}}{2 \cdot 1}$

Step 1.5.1.3

Apply the distributive property.

$y = \frac{- 14 \pm \sqrt{196 - 4 (- 4 x) - 4 \cdot 97}}{2 \cdot 1}$

Step 1.5.1.4

Multiply $- 4$ by $- 4$ .

$y = \frac{- 14 \pm \sqrt{196 + 16 x - 4 \cdot 97}}{2 \cdot 1}$

Step 1.5.1.5

Multiply $- 4$ by $97$ .

$y = \frac{- 14 \pm \sqrt{196 + 16 x - 388}}{2 \cdot 1}$

Step 1.5.1.6

Subtract $388$ from $196$ .

$y = \frac{- 14 \pm \sqrt{16 x - 192}}{2 \cdot 1}$

Step 1.5.1.7

Factor $16$ out of $16 x - 192$ .

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Step 1.5.1.7.1

Factor $16$ out of $16 x$ .

$y = \frac{- 14 \pm \sqrt{16 (x) - 192}}{2 \cdot 1}$

Step 1.5.1.7.2

Factor $16$ out of $- 192$ .

$y = \frac{- 14 \pm \sqrt{16 x + 16 \cdot - 12}}{2 \cdot 1}$

Step 1.5.1.7.3

Factor $16$ out of $16 x + 16 \cdot - 12$ .

$y = \frac{- 14 \pm \sqrt{16 (x - 12)}}{2 \cdot 1}$

Step 1.5.1.8

Rewrite $16 (x - 12)$ as ${(2^{2})}^{2} (x - 12)$ .

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Step 1.5.1.8.1

Rewrite $16$ as $4^{2}$ .

$y = \frac{- 14 \pm \sqrt{4^{2} (x - 12)}}{2 \cdot 1}$

Step 1.5.1.8.2

Rewrite $4$ as $2^{2}$ .

$y = \frac{- 14 \pm \sqrt{{(2^{2})}^{2} (x - 12)}}{2 \cdot 1}$

Step 1.5.1.9

Pull terms out from under the radical.

$y = \frac{- 14 \pm 2^{2} \sqrt{x - 12}}{2 \cdot 1}$

Step 1.5.1.10

Raise $2$ to the power of $2$ .

$y = \frac{- 14 \pm 4 \sqrt{x - 12}}{2 \cdot 1}$

Step 1.5.2

Multiply $2$ by $1$ .

$y = \frac{- 14 \pm 4 \sqrt{x - 12}}{2}$

Step 1.5.3

Simplify $\frac{- 14 \pm 4 \sqrt{x - 12}}{2}$ .

$y = - 7 \pm 2 \sqrt{x - 12}$

Step 1.5.4

Change the $\pm$ to $+$ .

$y = - 7 + 2 \sqrt{x - 12}$

Step 1.6

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

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Step 1.6.1

Simplify the numerator.

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Step 1.6.1.1

Raise $14$ to the power of $2$ .

$y = \frac{- 14 \pm \sqrt{196 - 4 \cdot 1 \cdot (- 4 x + 97)}}{2 \cdot 1}$

Step 1.6.1.2

Multiply $- 4$ by $1$ .

$y = \frac{- 14 \pm \sqrt{196 - 4 \cdot (- 4 x + 97)}}{2 \cdot 1}$

Step 1.6.1.3

Apply the distributive property.

$y = \frac{- 14 \pm \sqrt{196 - 4 (- 4 x) - 4 \cdot 97}}{2 \cdot 1}$

Step 1.6.1.4

Multiply $- 4$ by $- 4$ .

$y = \frac{- 14 \pm \sqrt{196 + 16 x - 4 \cdot 97}}{2 \cdot 1}$

Step 1.6.1.5

Multiply $- 4$ by $97$ .

$y = \frac{- 14 \pm \sqrt{196 + 16 x - 388}}{2 \cdot 1}$

Step 1.6.1.6

Subtract $388$ from $196$ .

$y = \frac{- 14 \pm \sqrt{16 x - 192}}{2 \cdot 1}$

Step 1.6.1.7

Factor $16$ out of $16 x - 192$ .

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Step 1.6.1.7.1

Factor $16$ out of $16 x$ .

$y = \frac{- 14 \pm \sqrt{16 (x) - 192}}{2 \cdot 1}$

Step 1.6.1.7.2

Factor $16$ out of $- 192$ .

$y = \frac{- 14 \pm \sqrt{16 x + 16 \cdot - 12}}{2 \cdot 1}$

Step 1.6.1.7.3

Factor $16$ out of $16 x + 16 \cdot - 12$ .

$y = \frac{- 14 \pm \sqrt{16 (x - 12)}}{2 \cdot 1}$

Step 1.6.1.8

Rewrite $16 (x - 12)$ as ${(2^{2})}^{2} (x - 12)$ .

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Step 1.6.1.8.1

Rewrite $16$ as $4^{2}$ .

$y = \frac{- 14 \pm \sqrt{4^{2} (x - 12)}}{2 \cdot 1}$

Step 1.6.1.8.2

Rewrite $4$ as $2^{2}$ .

$y = \frac{- 14 \pm \sqrt{{(2^{2})}^{2} (x - 12)}}{2 \cdot 1}$

Step 1.6.1.9

Pull terms out from under the radical.

$y = \frac{- 14 \pm 2^{2} \sqrt{x - 12}}{2 \cdot 1}$

Step 1.6.1.10

Raise $2$ to the power of $2$ .

$y = \frac{- 14 \pm 4 \sqrt{x - 12}}{2 \cdot 1}$

Step 1.6.2

Multiply $2$ by $1$ .

$y = \frac{- 14 \pm 4 \sqrt{x - 12}}{2}$

Step 1.6.3

Simplify $\frac{- 14 \pm 4 \sqrt{x - 12}}{2}$ .

$y = - 7 \pm 2 \sqrt{x - 12}$

Step 1.6.4

Change the $\pm$ to $-$ .

$y = - 7 - 2 \sqrt{x - 12}$

Step 1.7

The final answer is the combination of both solutions.

$y = - 7 + 2 \sqrt{x - 12}$

$y = - 7 - 2 \sqrt{x - 12}$

$y = - 7 + 2 \sqrt{x - 12}$

$y = - 7 - 2 \sqrt{x - 12}$

Step 2

To write a polynomial in standard form, simplify and then arrange the terms in descending order.

$y = a x^{2} + b x + c$

Step 3

The standard form is $- 7 + 2 \sqrt{x - 12}, - 7 - 2 \sqrt{x - 12}$ .

$y = - 7 + 2 \sqrt{x - 12}$

$y = - 7 - 2 \sqrt{x - 12}$

Step 4