Precalculus Examples

${(t - 4)}^{2} = {(t + 4)}^{2} + 32$

Step 1

Simplify ${(t + 4)}^{2} + 32$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite ${(t + 4)}^{2}$ as $(t + 4) (t + 4)$ .

${(t - 4)}^{2} = (t + 4) (t + 4) + 32$

Step 1.1.2

Expand $(t + 4) (t + 4)$ using the FOIL Method.

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Step 1.1.2.1

Apply the distributive property.

${(t - 4)}^{2} = t (t + 4) + 4 (t + 4) + 32$

Step 1.1.2.2

Apply the distributive property.

${(t - 4)}^{2} = t \cdot t + t \cdot 4 + 4 (t + 4) + 32$

Step 1.1.2.3

Apply the distributive property.

${(t - 4)}^{2} = t \cdot t + t \cdot 4 + 4 t + 4 \cdot 4 + 32$

Step 1.1.3

Simplify and combine like terms.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

Multiply $t$ by $t$ .

${(t - 4)}^{2} = t^{2} + t \cdot 4 + 4 t + 4 \cdot 4 + 32$

Step 1.1.3.1.2

Move $4$ to the left of $t$ .

${(t - 4)}^{2} = t^{2} + 4 \cdot t + 4 t + 4 \cdot 4 + 32$

Step 1.1.3.1.3

Multiply $4$ by $4$ .

${(t - 4)}^{2} = t^{2} + 4 t + 4 t + 16 + 32$

Step 1.1.3.2

Add $4 t$ and $4 t$ .

${(t - 4)}^{2} = t^{2} + 8 t + 16 + 32$

Step 1.2

Add $16$ and $32$ .

${(t - 4)}^{2} = t^{2} + 8 t + 48$

Step 2

Since $t$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$t^{2} + 8 t + 48 = {(t - 4)}^{2}$

Step 3

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

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Step 3.1

Rewrite ${(t - 4)}^{2}$ as $(t - 4) (t - 4)$ .

$t^{2} + 8 t + 48 = (t - 4) (t - 4)$

Step 3.2

Expand $(t - 4) (t - 4)$ using the FOIL Method.

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Step 3.2.1

Apply the distributive property.

$t^{2} + 8 t + 48 = t (t - 4) - 4 (t - 4)$

Step 3.2.2

Apply the distributive property.

$t^{2} + 8 t + 48 = t \cdot t + t \cdot - 4 - 4 (t - 4)$

Step 3.2.3

Apply the distributive property.

$t^{2} + 8 t + 48 = t \cdot t + t \cdot - 4 - 4 t - 4 \cdot - 4$

Step 3.3

Simplify and combine like terms.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Multiply $t$ by $t$ .

$t^{2} + 8 t + 48 = t^{2} + t \cdot - 4 - 4 t - 4 \cdot - 4$

Step 3.3.1.2

Move $- 4$ to the left of $t$ .

$t^{2} + 8 t + 48 = t^{2} - 4 \cdot t - 4 t - 4 \cdot - 4$

Step 3.3.1.3

Multiply $- 4$ by $- 4$ .

$t^{2} + 8 t + 48 = t^{2} - 4 t - 4 t + 16$

Step 3.3.2

Subtract $4 t$ from $- 4 t$ .

$t^{2} + 8 t + 48 = t^{2} - 8 t + 16$

Step 4

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

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Step 4.1

Subtract $t^{2}$ from both sides of the equation.

$t^{2} + 8 t + 48 - t^{2} = - 8 t + 16$

Step 4.2

Add $8 t$ to both sides of the equation.

$t^{2} + 8 t + 48 - t^{2} + 8 t = 16$

Step 4.3

Combine the opposite terms in $t^{2} + 8 t + 48 - t^{2} + 8 t$ .

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Step 4.3.1

Subtract $t^{2}$ from $t^{2}$ .

$8 t + 48 + 0 + 8 t = 16$

Step 4.3.2

Add $8 t + 48$ and $0$ .

$8 t + 48 + 8 t = 16$

Step 4.4

Add $8 t$ and $8 t$ .

$16 t + 48 = 16$

Step 5

Move all terms not containing $t$ to the right side of the equation.

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Step 5.1

Subtract $48$ from both sides of the equation.

$16 t = 16 - 48$

Step 5.2

Subtract $48$ from $16$ .

$16 t = - 32$

Step 6

Divide each term in $16 t = - 32$ by $16$ and simplify.

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Step 6.1

Divide each term in $16 t = - 32$ by $16$ .

$\frac{16 t}{16} = \frac{- 32}{16}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $16$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{16 t}{16} = \frac{- 32}{16}$

Step 6.2.1.2

Divide $t$ by $1$ .

$t = \frac{- 32}{16}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $- 32$ by $16$ .

$t = - 2$