Finite Math Examples

$x^{2} + y^{2} + 4 x - 4 y - 73 = 0$

Step 1

Use the quadratic formula to find the solutions.

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

Step 2

Substitute the values $a = 1$ , $b = - 4$ , and $c = x^{2} + 4 x - 73$ into the quadratic formula and solve for $y$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (1 \cdot (x^{2} + 4 x - 73))}}{2 \cdot 1}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Factor $- 4$ out of ${(- 4)}^{2} - 4 \cdot 1 \cdot (x^{2} + 4 x - 73)$ .

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Step 3.1.1.1

Factor $- 4$ out of ${(- 4)}^{2}$ .

$y = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 \cdot 1 \cdot (x^{2} + 4 x - 73)}}{2 \cdot 1}$

Step 3.1.1.2

Factor $- 4$ out of $- 4 \cdot 1 \cdot (x^{2} + 4 x - 73)$ .

$y = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 (1 \cdot (x^{2} + 4 x - 73))}}{2 \cdot 1}$

Step 3.1.1.3

Factor $- 4$ out of $- 4 \cdot - 4 - 4 (1 \cdot (x^{2} + 4 x - 73))$ .

$y = \frac{4 \pm \sqrt{- 4 (- 4 + 1 \cdot (x^{2} + 4 x - 73))}}{2 \cdot 1}$

Step 3.1.2

Multiply $x^{2} + 4 x - 73$ by $1$ .

$y = \frac{4 \pm \sqrt{- 4 (- 4 + x^{2} + 4 x - 73)}}{2 \cdot 1}$

Step 3.1.3

Subtract $73$ from $- 4$ .

$y = \frac{4 \pm \sqrt{- 4 (x^{2} + 4 x - 77)}}{2 \cdot 1}$

Step 3.1.4

Factor $x^{2} + 4 x - 77$ using the AC method.

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Step 3.1.4.1

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 $- 77$ and whose sum is $4$ .

$- 7, 11$

Step 3.1.4.2

Write the factored form using these integers.

$y = \frac{4 \pm \sqrt{- 4 ((x - 7) (x + 11))}}{2 \cdot 1}$

$y = \frac{4 \pm \sqrt{- 4 (x - 7) (x + 11)}}{2 \cdot 1}$

Step 3.1.5

Rewrite $- 4 (x - 7) (x + 11)$ as $2^{2} \cdot (- 1 (x - 7) (x + 11))$ .

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Step 3.1.5.1

Factor $4$ out of $- 4$ .

$y = \frac{4 \pm \sqrt{4 (- 1) (x - 7) (x + 11)}}{2 \cdot 1}$

Step 3.1.5.2

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

$y = \frac{4 \pm \sqrt{2^{2} \cdot (- 1 (x - 7) (x + 11))}}{2 \cdot 1}$

Step 3.1.5.3

Add parentheses.

$y = \frac{4 \pm \sqrt{2^{2} \cdot (- 1 ((x - 7) (x + 11)))}}{2 \cdot 1}$

Step 3.1.5.4

Add parentheses.

$y = \frac{4 \pm \sqrt{2^{2} \cdot (- 1 (x - 7) (x + 11))}}{2 \cdot 1}$

Step 3.1.6

Pull terms out from under the radical.

$y = \frac{4 \pm 2 \sqrt{- 1 (x - 7) (x + 11)}}{2 \cdot 1}$

Step 3.2

Multiply $2$ by $1$ .

$y = \frac{4 \pm 2 \sqrt{- 1 (x - 7) (x + 11)}}{2}$

Step 3.3

Simplify $\frac{4 \pm 2 \sqrt{- 1 (x - 7) (x + 11)}}{2}$ .

$y = 2 \pm \sqrt{- 1 (x - 7) (x + 11)}$

Step 4

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

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Factor $- 4$ out of ${(- 4)}^{2} - 4 \cdot 1 \cdot (x^{2} + 4 x - 73)$ .

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Step 4.1.1.1

Factor $- 4$ out of ${(- 4)}^{2}$ .

$y = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 \cdot 1 \cdot (x^{2} + 4 x - 73)}}{2 \cdot 1}$

Step 4.1.1.2

Factor $- 4$ out of $- 4 \cdot 1 \cdot (x^{2} + 4 x - 73)$ .

$y = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 (1 \cdot (x^{2} + 4 x - 73))}}{2 \cdot 1}$

Step 4.1.1.3

Factor $- 4$ out of $- 4 \cdot - 4 - 4 (1 \cdot (x^{2} + 4 x - 73))$ .

$y = \frac{4 \pm \sqrt{- 4 (- 4 + 1 \cdot (x^{2} + 4 x - 73))}}{2 \cdot 1}$

Step 4.1.2

Multiply $x^{2} + 4 x - 73$ by $1$ .

$y = \frac{4 \pm \sqrt{- 4 (- 4 + x^{2} + 4 x - 73)}}{2 \cdot 1}$

Step 4.1.3

Subtract $73$ from $- 4$ .

$y = \frac{4 \pm \sqrt{- 4 (x^{2} + 4 x - 77)}}{2 \cdot 1}$

Step 4.1.4

Factor $x^{2} + 4 x - 77$ using the AC method.

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Step 4.1.4.1

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 $- 77$ and whose sum is $4$ .

$- 7, 11$

Step 4.1.4.2

Write the factored form using these integers.

$y = \frac{4 \pm \sqrt{- 4 ((x - 7) (x + 11))}}{2 \cdot 1}$

$y = \frac{4 \pm \sqrt{- 4 (x - 7) (x + 11)}}{2 \cdot 1}$

Step 4.1.5

Rewrite $- 4 (x - 7) (x + 11)$ as $2^{2} \cdot (- 1 (x - 7) (x + 11))$ .

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Step 4.1.5.1

Factor $4$ out of $- 4$ .

$y = \frac{4 \pm \sqrt{4 (- 1) (x - 7) (x + 11)}}{2 \cdot 1}$

Step 4.1.5.2

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

$y = \frac{4 \pm \sqrt{2^{2} \cdot (- 1 (x - 7) (x + 11))}}{2 \cdot 1}$

Step 4.1.5.3

Add parentheses.

$y = \frac{4 \pm \sqrt{2^{2} \cdot (- 1 ((x - 7) (x + 11)))}}{2 \cdot 1}$

Step 4.1.5.4

Add parentheses.

$y = \frac{4 \pm \sqrt{2^{2} \cdot (- 1 (x - 7) (x + 11))}}{2 \cdot 1}$

Step 4.1.6

Pull terms out from under the radical.

$y = \frac{4 \pm 2 \sqrt{- 1 (x - 7) (x + 11)}}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

$y = \frac{4 \pm 2 \sqrt{- 1 (x - 7) (x + 11)}}{2}$

Step 4.3

Simplify $\frac{4 \pm 2 \sqrt{- 1 (x - 7) (x + 11)}}{2}$ .

$y = 2 \pm \sqrt{- 1 (x - 7) (x + 11)}$

Step 4.4

Change the $\pm$ to $+$ .

$y = 2 + \sqrt{- 1 (x - 7) (x + 11)}$

Step 5

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

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Factor $- 4$ out of ${(- 4)}^{2} - 4 \cdot 1 \cdot (x^{2} + 4 x - 73)$ .

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Step 5.1.1.1

Factor $- 4$ out of ${(- 4)}^{2}$ .

$y = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 \cdot 1 \cdot (x^{2} + 4 x - 73)}}{2 \cdot 1}$

Step 5.1.1.2

Factor $- 4$ out of $- 4 \cdot 1 \cdot (x^{2} + 4 x - 73)$ .

$y = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 (1 \cdot (x^{2} + 4 x - 73))}}{2 \cdot 1}$

Step 5.1.1.3

Factor $- 4$ out of $- 4 \cdot - 4 - 4 (1 \cdot (x^{2} + 4 x - 73))$ .

$y = \frac{4 \pm \sqrt{- 4 (- 4 + 1 \cdot (x^{2} + 4 x - 73))}}{2 \cdot 1}$

Step 5.1.2

Multiply $x^{2} + 4 x - 73$ by $1$ .

$y = \frac{4 \pm \sqrt{- 4 (- 4 + x^{2} + 4 x - 73)}}{2 \cdot 1}$

Step 5.1.3

Subtract $73$ from $- 4$ .

$y = \frac{4 \pm \sqrt{- 4 (x^{2} + 4 x - 77)}}{2 \cdot 1}$

Step 5.1.4

Factor $x^{2} + 4 x - 77$ using the AC method.

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Step 5.1.4.1

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 $- 77$ and whose sum is $4$ .

$- 7, 11$

Step 5.1.4.2

Write the factored form using these integers.

$y = \frac{4 \pm \sqrt{- 4 ((x - 7) (x + 11))}}{2 \cdot 1}$

$y = \frac{4 \pm \sqrt{- 4 (x - 7) (x + 11)}}{2 \cdot 1}$

Step 5.1.5

Rewrite $- 4 (x - 7) (x + 11)$ as $2^{2} \cdot (- 1 (x - 7) (x + 11))$ .

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Step 5.1.5.1

Factor $4$ out of $- 4$ .

$y = \frac{4 \pm \sqrt{4 (- 1) (x - 7) (x + 11)}}{2 \cdot 1}$

Step 5.1.5.2

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

$y = \frac{4 \pm \sqrt{2^{2} \cdot (- 1 (x - 7) (x + 11))}}{2 \cdot 1}$

Step 5.1.5.3

Add parentheses.

$y = \frac{4 \pm \sqrt{2^{2} \cdot (- 1 ((x - 7) (x + 11)))}}{2 \cdot 1}$

Step 5.1.5.4

Add parentheses.

$y = \frac{4 \pm \sqrt{2^{2} \cdot (- 1 (x - 7) (x + 11))}}{2 \cdot 1}$

Step 5.1.6

Pull terms out from under the radical.

$y = \frac{4 \pm 2 \sqrt{- 1 (x - 7) (x + 11)}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$y = \frac{4 \pm 2 \sqrt{- 1 (x - 7) (x + 11)}}{2}$

Step 5.3

Simplify $\frac{4 \pm 2 \sqrt{- 1 (x - 7) (x + 11)}}{2}$ .

$y = 2 \pm \sqrt{- 1 (x - 7) (x + 11)}$

Step 5.4

Change the $\pm$ to $-$ .

$y = 2 - \sqrt{- 1 (x - 7) (x + 11)}$

Step 6

The final answer is the combination of both solutions.

$y = 2 + \sqrt{- 1 (x - 7) (x + 11)}$

$y = 2 - \sqrt{- 1 (x - 7) (x + 11)}$

Step 7

Set the radicand in $\sqrt{- 1 (x - 7) (x + 11)}$ greater than or equal to $0$ to find where the expression is defined.

$- 1 (x - 7) (x + 11) \geq 0$

Step 8

Solve for $x$ .

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Step 8.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 7 = 0$

$x + 11 = 0$

Step 8.2

Set $x - 7$ equal to $0$ and solve for $x$ .

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Step 8.2.1

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Step 8.2.2

Add $7$ to both sides of the equation.

$x = 7$

Step 8.3

Set $x + 11$ equal to $0$ and solve for $x$ .

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Step 8.3.1

Set $x + 11$ equal to $0$ .

$x + 11 = 0$

Step 8.3.2

Subtract $11$ from both sides of the equation.

$x = - 11$

Step 8.4

The final solution is all the values that make $- 1 (x - 7) (x + 11) \geq 0$ true.

$x = 7, - 11$

Step 8.5

Use each root to create test intervals.

$x < - 11$

$- 11 < x < 7$

$x > 7$

Step 8.6

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 8.6.1

Test a value on the interval $x < - 11$ to see if it makes the inequality true.

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Step 8.6.1.1

Choose a value on the interval $x < - 11$ and see if this value makes the original inequality true.

$x = - 14$

Step 8.6.1.2

Replace $x$ with $- 14$ in the original inequality.

$- 1 ((- 14) - 7) ((- 14) + 11) \geq 0$

Step 8.6.1.3

The left side $- 63$ is less than the right side $0$ , which means that the given statement is false.

False

Step 8.6.2

Test a value on the interval $- 11 < x < 7$ to see if it makes the inequality true.

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Step 8.6.2.1

Choose a value on the interval $- 11 < x < 7$ and see if this value makes the original inequality true.

$x = 0$

Step 8.6.2.2

Replace $x$ with $0$ in the original inequality.

$- 1 ((0) - 7) ((0) + 11) \geq 0$

Step 8.6.2.3

The left side $77$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 8.6.3

Test a value on the interval $x > 7$ to see if it makes the inequality true.

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Step 8.6.3.1

Choose a value on the interval $x > 7$ and see if this value makes the original inequality true.

$x = 10$

Step 8.6.3.2

Replace $x$ with $10$ in the original inequality.

$- 1 ((10) - 7) ((10) + 11) \geq 0$

Step 8.6.3.3

The left side $- 63$ is less than the right side $0$ , which means that the given statement is false.

False

Step 8.6.4

Compare the intervals to determine which ones satisfy the original inequality.

$x < - 11$ False

$- 11 < x < 7$ True

$x > 7$ False

$x < - 11$ False

$- 11 < x < 7$ True

$x > 7$ False

Step 8.7

The solution consists of all of the true intervals.

$- 11 \leq x \leq 7$

Step 9

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$[- 11, 7]$

Set-Builder Notation:

${x | - 11 \leq x \leq 7}$

Step 10

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$[- 7, 11]$

Set-Builder Notation:

${y | - 7 \leq y \leq 11}$

Step 11

Determine the domain and range.

Domain: $[- 11, 7], {x | - 11 \leq x \leq 7}$

Range: $[- 7, 11], {y | - 7 \leq y \leq 11}$

Step 12