Precalculus Examples

$y = x^{2} - 3 x - 4$ , $y = 5 x + 11$

Step 1

Eliminate the equal sides of each equation and combine.

$x^{2} - 3 x - 4 = 5 x + 11$

Step 2

Solve $x^{2} - 3 x - 4 = 5 x + 11$ for $x$ .

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Step 2.1

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

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Step 2.1.1

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

$x^{2} - 3 x - 4 - 5 x = 11$

Step 2.1.2

Subtract $5 x$ from $- 3 x$ .

$x^{2} - 8 x - 4 = 11$

Step 2.2

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

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Step 2.2.1

Subtract $11$ from both sides of the equation.

$x^{2} - 8 x - 4 - 11 = 0$

Step 2.2.2

Subtract $11$ from $- 4$ .

$x^{2} - 8 x - 15 = 0$

Step 2.3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a} + y = 5 x + 11$

Step 2.4

Substitute the values $a = 1$ , $b = - 8$ , and $c = - 15$ into the quadratic formula and solve for $x$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (1 \cdot - 15)}}{2 \cdot 1} + y = 5 x + 11$

Step 2.5

Simplify.

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot - 15}}{2 \cdot 1}$

Step 2.5.1.2

Multiply $- 4 \cdot 1 \cdot - 15$ .

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Step 2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot - 15}}{2 \cdot 1}$

Step 2.5.1.2.2

Multiply $- 4$ by $- 15$ .

$x = \frac{8 \pm \sqrt{64 + 60}}{2 \cdot 1}$

Step 2.5.1.3

Add $64$ and $60$ .

$x = \frac{8 \pm \sqrt{124}}{2 \cdot 1}$

Step 2.5.1.4

Rewrite $124$ as $2^{2} \cdot 31$ .

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Step 2.5.1.4.1

Factor $4$ out of $124$ .

$x = \frac{8 \pm \sqrt{4 (31)}}{2 \cdot 1}$

Step 2.5.1.4.2

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

$x = \frac{8 \pm \sqrt{2^{2} \cdot 31}}{2 \cdot 1}$

Step 2.5.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 2 \sqrt{31}}{2 \cdot 1}$

Step 2.5.2

Multiply $2$ by $1$ .

$x = \frac{8 \pm 2 \sqrt{31}}{2}$

Step 2.5.3

Simplify $\frac{8 \pm 2 \sqrt{31}}{2}$ .

$x = 4 \pm \sqrt{31}$

Step 2.6

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

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Step 2.6.1

Simplify the numerator.

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Step 2.6.1.1

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot - 15}}{2 \cdot 1}$

Step 2.6.1.2

Multiply $- 4 \cdot 1 \cdot - 15$ .

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Step 2.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot - 15}}{2 \cdot 1}$

Step 2.6.1.2.2

Multiply $- 4$ by $- 15$ .

$x = \frac{8 \pm \sqrt{64 + 60}}{2 \cdot 1}$

Step 2.6.1.3

Add $64$ and $60$ .

$x = \frac{8 \pm \sqrt{124}}{2 \cdot 1}$

Step 2.6.1.4

Rewrite $124$ as $2^{2} \cdot 31$ .

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Step 2.6.1.4.1

Factor $4$ out of $124$ .

$x = \frac{8 \pm \sqrt{4 (31)}}{2 \cdot 1}$

Step 2.6.1.4.2

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

$x = \frac{8 \pm \sqrt{2^{2} \cdot 31}}{2 \cdot 1}$

Step 2.6.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 2 \sqrt{31}}{2 \cdot 1}$

Step 2.6.2

Multiply $2$ by $1$ .

$x = \frac{8 \pm 2 \sqrt{31}}{2}$

Step 2.6.3

Simplify $\frac{8 \pm 2 \sqrt{31}}{2}$ .

$x = 4 \pm \sqrt{31}$

Step 2.6.4

Change the $\pm$ to $+$ .

$x = 4 + \sqrt{31}$

Step 2.7

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

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Step 2.7.1

Simplify the numerator.

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Step 2.7.1.1

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot - 15}}{2 \cdot 1}$

Step 2.7.1.2

Multiply $- 4 \cdot 1 \cdot - 15$ .

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Step 2.7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot - 15}}{2 \cdot 1}$

Step 2.7.1.2.2

Multiply $- 4$ by $- 15$ .

$x = \frac{8 \pm \sqrt{64 + 60}}{2 \cdot 1}$

Step 2.7.1.3

Add $64$ and $60$ .

$x = \frac{8 \pm \sqrt{124}}{2 \cdot 1}$

Step 2.7.1.4

Rewrite $124$ as $2^{2} \cdot 31$ .

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Step 2.7.1.4.1

Factor $4$ out of $124$ .

$x = \frac{8 \pm \sqrt{4 (31)}}{2 \cdot 1}$

Step 2.7.1.4.2

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

$x = \frac{8 \pm \sqrt{2^{2} \cdot 31}}{2 \cdot 1}$

Step 2.7.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 2 \sqrt{31}}{2 \cdot 1}$

Step 2.7.2

Multiply $2$ by $1$ .

$x = \frac{8 \pm 2 \sqrt{31}}{2}$

Step 2.7.3

Simplify $\frac{8 \pm 2 \sqrt{31}}{2}$ .

$x = 4 \pm \sqrt{31}$

Step 2.7.4

Change the $\pm$ to $-$ .

$x = 4 - \sqrt{31}$

Step 2.8

The final answer is the combination of both solutions.

$x = 4 + \sqrt{31}, 4 - \sqrt{31}$

Step 3

Evaluate $y$ when $x = 4 + \sqrt{31}$ .

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Step 3.1

Substitute $4 + \sqrt{31}$ for $x$ .

$y = 5 (4 + \sqrt{31}) + 11$

Step 3.2

Simplify $5 (4 + \sqrt{31}) + 11$ .

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Apply the distributive property.

$y = 5 \cdot 4 + 5 \sqrt{31} + 11$

Step 3.2.1.2

Multiply $5$ by $4$ .

$y = 20 + 5 \sqrt{31} + 11$

Step 3.2.2

Add $20$ and $11$ .

$y = 31 + 5 \sqrt{31}$

Step 4

Evaluate $y$ when $x = 4 - \sqrt{31}$ .

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Step 4.1

Substitute $4 - \sqrt{31}$ for $x$ .

$y = 5 (4 - \sqrt{31}) + 11$

Step 4.2

Simplify $5 (4 - \sqrt{31}) + 11$ .

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Apply the distributive property.

$y = 5 \cdot 4 + 5 (- \sqrt{31}) + 11$

Step 4.2.1.2

Multiply $5$ by $4$ .

$y = 20 + 5 (- \sqrt{31}) + 11$

Step 4.2.1.3

Multiply $- 1$ by $5$ .

$y = 20 - 5 \sqrt{31} + 11$

Step 4.2.2

Add $20$ and $11$ .

$y = 31 - 5 \sqrt{31}$

Step 5

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(4 + \sqrt{31}, 31 + 5 \sqrt{31})$

$(4 - \sqrt{31}, 31 - 5 \sqrt{31})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(4 + \sqrt{31}, 31 + 5 \sqrt{31}), (4 - \sqrt{31}, 31 - 5 \sqrt{31})$

Equation Form:

$x = 4 + \sqrt{31}, y = 31 + 5 \sqrt{31}$

$x = 4 - \sqrt{31}, y = 31 - 5 \sqrt{31}$

Step 7