Precalculus Examples

$16 {(x + 1)}^{2} + 8 (x + 1) + 1 = 0$

Factor the left side of the equation.

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Let $u = x + 1$ . Substitute $u$ for all occurrences of $x + 1$ .

$16 u^{2} + 8 u + 1 = 0$

Factor using the perfect square rule.

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Rewrite $16 u^{2}$ as ${(4 u)}^{2}$ .

${(4 u)}^{2} + 8 u + 1 = 0$

Rewrite $1$ as $1^{2}$ .

${(4 u)}^{2} + 8 u + 1^{2} = 0$

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

$8 u = 2 \cdot (4 u) \cdot 1$

Rewrite the polynomial.

${(4 u)}^{2} + 2 \cdot (4 u) \cdot 1 + 1^{2} = 0$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = 4 u$ and $b = 1$ .

${(4 u + 1)}^{2} = 0$

Replace all occurrences of $u$ with $x + 1$ .

${(4 (x + 1) + 1)}^{2} = 0$

Set the $4 (x + 1) + 1$ equal to $0$ .

$4 (x + 1) + 1 = 0$

Solve for $x$ .

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Simplify $4 (x + 1) + 1$ .

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Simplify each term.

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Apply the distributive property.

$4 x + 4 \cdot 1 + 1 = 0$

Multiply $4$ by $1$ .

$4 x + 4 + 1 = 0$

Add $4$ and $1$ .

$4 x + 5 = 0$

Subtract $5$ from both sides of the equation.

$4 x = - 5$

Divide each term in $4 x = - 5$ by $4$ and simplify.

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Divide each term in $4 x = - 5$ by $4$ .

$\frac{4 x}{4} = \frac{- 5}{4}$

Simplify the left side.

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Cancel the common factor of $4$ .

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Cancel the common factor.

$\frac{4 x}{4} = \frac{- 5}{4}$

Divide $x$ by $1$ .

$x = \frac{- 5}{4}$

Simplify the right side.

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Move the negative in front of the fraction.

$x = - \frac{5}{4}$

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{5}{4}$

Decimal Form:

$x = - 1.25$

Mixed Number Form:

$x = - 1 \frac{1}{4}$