Finite Math Examples

$x^{2} - 16 x + y^{2} - 10 y + 64 = 0$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$x^{2} - 16 x + {(0)}^{2} - 10 \cdot 0 + 64 = 0$

Step 1.2

Solve the equation.

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Step 1.2.1

Simplify $x^{2} - 16 x + {(0)}^{2} - 10 \cdot 0 + 64$ .

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Step 1.2.1.1

Simplify each term.

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Step 1.2.1.1.1

Raising $0$ to any positive power yields $0$ .

$x^{2} - 16 x + 0 - 10 \cdot 0 + 64 = 0$

Step 1.2.1.1.2

Multiply $- 10$ by $0$ .

$x^{2} - 16 x + 0 + 0 + 64 = 0$

Step 1.2.1.2

Combine the opposite terms in $x^{2} - 16 x + 0 + 0 + 64$ .

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Step 1.2.1.2.1

Add $x^{2} - 16 x$ and $0$ .

$x^{2} - 16 x + 0 + 64 = 0$

Step 1.2.1.2.2

Add $x^{2} - 16 x$ and $0$ .

$x^{2} - 16 x + 64 = 0$

Step 1.2.2

Factor using the perfect square rule.

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Step 1.2.2.1

Rewrite $64$ as $8^{2}$ .

$x^{2} - 16 x + 8^{2} = 0$

Step 1.2.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$16 x = 2 \cdot x \cdot 8$

Step 1.2.2.3

Rewrite the polynomial.

$x^{2} - 2 \cdot x \cdot 8 + 8^{2} = 0$

Step 1.2.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 8$ .

${(x - 8)}^{2} = 0$

Step 1.2.3

Set the $x - 8$ equal to $0$ .

$x - 8 = 0$

Step 1.2.4

Add $8$ to both sides of the equation.

$x = 8$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(8, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

${(0)}^{2} - 16 \cdot 0 + y^{2} - 10 y + 64 = 0$

Step 2.2

Solve the equation.

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Step 2.2.1

Simplify ${(0)}^{2} - 16 \cdot 0 + y^{2} - 10 y + 64$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Raising $0$ to any positive power yields $0$ .

$0 - 16 \cdot 0 + y^{2} - 10 y + 64 = 0$

Step 2.2.1.1.2

Multiply $- 16$ by $0$ .

$0 + 0 + y^{2} - 10 y + 64 = 0$

Step 2.2.1.2

Combine the opposite terms in $0 + 0 + y^{2} - 10 y + 64$ .

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Step 2.2.1.2.1

Add $0$ and $0$ .

$0 + y^{2} - 10 y + 64 = 0$

Step 2.2.1.2.2

Add $0$ and $y^{2}$ .

$y^{2} - 10 y + 64 = 0$

Step 2.2.2

Use the quadratic formula to find the solutions.

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

Step 2.2.3

Substitute the values $a = 1$ , $b = - 10$ , and $c = 64$ into the quadratic formula and solve for $y$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (1 \cdot 64)}}{2 \cdot 1}$

Step 2.2.4

Simplify.

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Step 2.2.4.1

Simplify the numerator.

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Step 2.2.4.1.1

Raise $- 10$ to the power of $2$ .

$y = \frac{10 \pm \sqrt{100 - 4 \cdot 1 \cdot 64}}{2 \cdot 1}$

Step 2.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 64$ .

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Step 2.2.4.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{10 \pm \sqrt{100 - 4 \cdot 64}}{2 \cdot 1}$

Step 2.2.4.1.2.2

Multiply $- 4$ by $64$ .

$y = \frac{10 \pm \sqrt{100 - 256}}{2 \cdot 1}$

Step 2.2.4.1.3

Subtract $256$ from $100$ .

$y = \frac{10 \pm \sqrt{- 156}}{2 \cdot 1}$

Step 2.2.4.1.4

Rewrite $- 156$ as $- 1 (156)$ .

$y = \frac{10 \pm \sqrt{- 1 \cdot 156}}{2 \cdot 1}$

Step 2.2.4.1.5

Rewrite $\sqrt{- 1 (156)}$ as $\sqrt{- 1} \cdot \sqrt{156}$ .

$y = \frac{10 \pm \sqrt{- 1} \cdot \sqrt{156}}{2 \cdot 1}$

Step 2.2.4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$y = \frac{10 \pm i \cdot \sqrt{156}}{2 \cdot 1}$

Step 2.2.4.1.7

Rewrite $156$ as $2^{2} \cdot 39$ .

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Step 2.2.4.1.7.1

Factor $4$ out of $156$ .

$y = \frac{10 \pm i \cdot \sqrt{4 (39)}}{2 \cdot 1}$

Step 2.2.4.1.7.2

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

$y = \frac{10 \pm i \cdot \sqrt{2^{2} \cdot 39}}{2 \cdot 1}$

Step 2.2.4.1.8

Pull terms out from under the radical.

$y = \frac{10 \pm i \cdot (2 \sqrt{39})}{2 \cdot 1}$

Step 2.2.4.1.9

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

$y = \frac{10 \pm 2 i \sqrt{39}}{2 \cdot 1}$

Step 2.2.4.2

Multiply $2$ by $1$ .

$y = \frac{10 \pm 2 i \sqrt{39}}{2}$

Step 2.2.4.3

Simplify $\frac{10 \pm 2 i \sqrt{39}}{2}$ .

$y = 5 \pm i \sqrt{39}$

Step 2.2.5

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

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Step 2.2.5.1

Simplify the numerator.

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Step 2.2.5.1.1

Raise $- 10$ to the power of $2$ .

$y = \frac{10 \pm \sqrt{100 - 4 \cdot 1 \cdot 64}}{2 \cdot 1}$

Step 2.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 64$ .

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Step 2.2.5.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{10 \pm \sqrt{100 - 4 \cdot 64}}{2 \cdot 1}$

Step 2.2.5.1.2.2

Multiply $- 4$ by $64$ .

$y = \frac{10 \pm \sqrt{100 - 256}}{2 \cdot 1}$

Step 2.2.5.1.3

Subtract $256$ from $100$ .

$y = \frac{10 \pm \sqrt{- 156}}{2 \cdot 1}$

Step 2.2.5.1.4

Rewrite $- 156$ as $- 1 (156)$ .

$y = \frac{10 \pm \sqrt{- 1 \cdot 156}}{2 \cdot 1}$

Step 2.2.5.1.5

Rewrite $\sqrt{- 1 (156)}$ as $\sqrt{- 1} \cdot \sqrt{156}$ .

$y = \frac{10 \pm \sqrt{- 1} \cdot \sqrt{156}}{2 \cdot 1}$

Step 2.2.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$y = \frac{10 \pm i \cdot \sqrt{156}}{2 \cdot 1}$

Step 2.2.5.1.7

Rewrite $156$ as $2^{2} \cdot 39$ .

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Step 2.2.5.1.7.1

Factor $4$ out of $156$ .

$y = \frac{10 \pm i \cdot \sqrt{4 (39)}}{2 \cdot 1}$

Step 2.2.5.1.7.2

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

$y = \frac{10 \pm i \cdot \sqrt{2^{2} \cdot 39}}{2 \cdot 1}$

Step 2.2.5.1.8

Pull terms out from under the radical.

$y = \frac{10 \pm i \cdot (2 \sqrt{39})}{2 \cdot 1}$

Step 2.2.5.1.9

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

$y = \frac{10 \pm 2 i \sqrt{39}}{2 \cdot 1}$

Step 2.2.5.2

Multiply $2$ by $1$ .

$y = \frac{10 \pm 2 i \sqrt{39}}{2}$

Step 2.2.5.3

Simplify $\frac{10 \pm 2 i \sqrt{39}}{2}$ .

$y = 5 \pm i \sqrt{39}$

Step 2.2.5.4

Change the $\pm$ to $+$ .

$y = 5 + i \sqrt{39}$

Step 2.2.6

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

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Step 2.2.6.1

Simplify the numerator.

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Step 2.2.6.1.1

Raise $- 10$ to the power of $2$ .

$y = \frac{10 \pm \sqrt{100 - 4 \cdot 1 \cdot 64}}{2 \cdot 1}$

Step 2.2.6.1.2

Multiply $- 4 \cdot 1 \cdot 64$ .

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Step 2.2.6.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{10 \pm \sqrt{100 - 4 \cdot 64}}{2 \cdot 1}$

Step 2.2.6.1.2.2

Multiply $- 4$ by $64$ .

$y = \frac{10 \pm \sqrt{100 - 256}}{2 \cdot 1}$

Step 2.2.6.1.3

Subtract $256$ from $100$ .

$y = \frac{10 \pm \sqrt{- 156}}{2 \cdot 1}$

Step 2.2.6.1.4

Rewrite $- 156$ as $- 1 (156)$ .

$y = \frac{10 \pm \sqrt{- 1 \cdot 156}}{2 \cdot 1}$

Step 2.2.6.1.5

Rewrite $\sqrt{- 1 (156)}$ as $\sqrt{- 1} \cdot \sqrt{156}$ .

$y = \frac{10 \pm \sqrt{- 1} \cdot \sqrt{156}}{2 \cdot 1}$

Step 2.2.6.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$y = \frac{10 \pm i \cdot \sqrt{156}}{2 \cdot 1}$

Step 2.2.6.1.7

Rewrite $156$ as $2^{2} \cdot 39$ .

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Step 2.2.6.1.7.1

Factor $4$ out of $156$ .

$y = \frac{10 \pm i \cdot \sqrt{4 (39)}}{2 \cdot 1}$

Step 2.2.6.1.7.2

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

$y = \frac{10 \pm i \cdot \sqrt{2^{2} \cdot 39}}{2 \cdot 1}$

Step 2.2.6.1.8

Pull terms out from under the radical.

$y = \frac{10 \pm i \cdot (2 \sqrt{39})}{2 \cdot 1}$

Step 2.2.6.1.9

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

$y = \frac{10 \pm 2 i \sqrt{39}}{2 \cdot 1}$

Step 2.2.6.2

Multiply $2$ by $1$ .

$y = \frac{10 \pm 2 i \sqrt{39}}{2}$

Step 2.2.6.3

Simplify $\frac{10 \pm 2 i \sqrt{39}}{2}$ .

$y = 5 \pm i \sqrt{39}$

Step 2.2.6.4

Change the $\pm$ to $-$ .

$y = 5 - i \sqrt{39}$

Step 2.2.7

The final answer is the combination of both solutions.

$y = 5 + i \sqrt{39}, 5 - i \sqrt{39}$

Step 2.3

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

y-intercept(s): $None$

Step 3

List the intersections.

x-intercept(s): $(8, 0)$

y-intercept(s): $None$

Step 4