Calculus Examples

$y = x^{2}$ , $y = {(x - 8)}^{2}$ , $y = 32$

Step 1

Solve by substitution to find the intersection between the curves.

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Step 1.1

Eliminate the equal sides of each equation and combine.

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

Step 1.2

Solve $x^{2} = {(x - 8)}^{2}$ for $x$ .

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Step 1.2.1

Since the exponents are equal, the bases of the exponents on both sides of the equation must be equal.

$| x | = | x - 8 | + y = {(x - 8)}^{2}$

$y = 32$

Step 1.2.2

Solve for $x$ .

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Step 1.2.2.1

Rewrite the absolute value equation as four equations without absolute value bars.

$x = x - 8$

$x = - (x - 8)$

$- x = x - 8$

$- x = - (x - 8) + y = {(x - 8)}^{2}$

$y = 32$

Step 1.2.2.2

After simplifying, there are only two unique equations to be solved.

$x = x - 8$

$x = - (x - 8) + y = {(x - 8)}^{2}$

$y = 32$

Step 1.2.2.3

Solve $x = x - 8$ for $x$ .

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Step 1.2.2.3.1

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

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Step 1.2.2.3.1.1

Subtract $x$ from both sides of the equation.

$x - x = - 8 + y = {(x - 8)}^{2}$

$y = 32$

Step 1.2.2.3.1.2

Subtract $x$ from $x$ .

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

$y = 32$

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

$y = 32$

Step 1.2.2.3.2

Since $0 \neq - 8$ , there are no solutions.

No $s o l u t i o n + y = {(x - 8)}^{2}$

$y = 32$

No $s o l u t i o n + y = {(x - 8)}^{2}$

$y = 32$

Step 1.2.2.4

Solve $x = - (x - 8)$ for $x$ .

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Step 1.2.2.4.1

Simplify $- (x - 8)$ .

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Step 1.2.2.4.1.1

Apply the distributive property.

$x = - x + 8 + y = {(x - 8)}^{2}$

$y = 32$

Step 1.2.2.4.1.2

Multiply $- 1$ by $- 8$ .

$x = - x + 8 + y = {(x - 8)}^{2}$

$y = 32$

$x = - x + 8 + y = {(x - 8)}^{2}$

$y = 32$

Step 1.2.2.4.2

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

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Step 1.2.2.4.2.1

Add $x$ to both sides of the equation.

$x + x = 8 + y = {(x - 8)}^{2}$

$y = 32$

Step 1.2.2.4.2.2

Add $x$ and $x$ .

$2 x = 8 + y = {(x - 8)}^{2}$

$y = 32$

$2 x = 8 + y = {(x - 8)}^{2}$

$y = 32$

Step 1.2.2.4.3

Divide each term in $2 x = 8$ by $2$ and simplify.

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Step 1.2.2.4.3.1

Divide each term in $2 x = 8$ by $2$ .

$\frac{2 x}{2} = \frac{8}{2} + y = {(x - 8)}^{2}$

$y = 32$

Step 1.2.2.4.3.2

Simplify the left side.

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Step 1.2.2.4.3.2.1

Cancel the common factor of $2$ .

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Step 1.2.2.4.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{8}{2} + y = {(x - 8)}^{2}$

$y = 32$

Step 1.2.2.4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{8}{2} + y = {(x - 8)}^{2}$

$y = 32$

$x = \frac{8}{2} + y = {(x - 8)}^{2}$

$y = 32$

$x = \frac{8}{2} + y = {(x - 8)}^{2}$

$y = 32$

Step 1.2.2.4.3.3

Simplify the right side.

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Step 1.2.2.4.3.3.1

Divide $8$ by $2$ .

$x = 4 + y = {(x - 8)}^{2}$

$y = 32$

$x = 4 + y = {(x - 8)}^{2}$

$y = 32$

$x = 4 + y = {(x - 8)}^{2}$

$y = 32$

$x = 4 + y = {(x - 8)}^{2}$

$y = 32$

Step 1.2.2.5

List all of the solutions.

$x = 4 + y = {(x - 8)}^{2}$

$y = 32$

$x = 4 + y = {(x - 8)}^{2}$

$y = 32$

$x = 4 + y = {(x - 8)}^{2}$

$y = 32$

Step 1.3

Evaluate $y$ when $x = 4$ .

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Step 1.3.1

Substitute $4$ for $x$ .

$y = {((4) - 8)}^{2} y = 32$

Step 1.3.2

Substitute $4$ for $x$ in $y = {((4) - 8)}^{2}$ and solve for $y$ .

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Step 1.3.2.1

Remove parentheses.

$y = {(4 - 8)}^{2}$

Step 1.3.2.2

Remove parentheses.

$y = {((4) - 8)}^{2}$

Step 1.3.2.3

Simplify ${((4) - 8)}^{2}$ .

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Step 1.3.2.3.1

Subtract $8$ from $4$ .

$y = {(- 4)}^{2}$

Step 1.3.2.3.2

Raise $- 4$ to the power of $2$ .

$y = 16$

Step 1.4

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

$(4, 16)$

Step 2

Simplify ${(x - 8)}^{2}$ .

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Step 2.1

Rewrite ${(x - 8)}^{2}$ as $(x - 8) (x - 8)$ .

$y = (x - 8) (x - 8)$

Step 2.2

Expand $(x - 8) (x - 8)$ using the FOIL Method.

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Step 2.2.1

Apply the distributive property.

$y = x (x - 8) - 8 (x - 8)$

Step 2.2.2

Apply the distributive property.

$y = x \cdot x + x \cdot - 8 - 8 (x - 8)$

Step 2.2.3

Apply the distributive property.

$y = x \cdot x + x \cdot - 8 - 8 x - 8 \cdot - 8$

Step 2.3

Simplify and combine like terms.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Multiply $x$ by $x$ .

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

Step 2.3.1.2

Move $- 8$ to the left of $x$ .

$y = x^{2} - 8 \cdot x - 8 x - 8 \cdot - 8$

Step 2.3.1.3

Multiply $- 8$ by $- 8$ .

$y = x^{2} - 8 x - 8 x + 64$

Step 2.3.2

Subtract $8 x$ from $- 8 x$ .

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