Algebra Examples

$x^{2} + 10 x - 8 = {(x - h)}^{2} - 33$

Step 1

Rewrite the equation as ${(x - h)}^{2} - 33 = x^{2} + 10 x - 8$ .

${(x - h)}^{2} - 33 = x^{2} + 10 x - 8$

Step 2

Add $33$ to both sides of the equation.

${(x - h)}^{2} = x^{2} + 10 x - 8 + 33$

Step 3

Add $- 8$ and $33$ .

${(x - h)}^{2} = x^{2} + 10 x + 25$

Step 4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x - h = \pm \sqrt{x^{2} + 10 x + 25}$

Step 5

Simplify $\pm \sqrt{x^{2} + 10 x + 25}$ .

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Step 5.1

Factor using the perfect square rule.

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Step 5.1.1

Rewrite $25$ as $5^{2}$ .

$x - h = \pm \sqrt{x^{2} + 10 x + 5^{2}}$

Step 5.1.2

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

$10 x = 2 \cdot x \cdot 5$

Step 5.1.3

Rewrite the polynomial.

$x - h = \pm \sqrt{x^{2} + 2 \cdot x \cdot 5 + 5^{2}}$

Step 5.1.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 5$ .

$x - h = \pm \sqrt{{(x + 5)}^{2}}$

Step 5.2

Pull terms out from under the radical, assuming positive real numbers.

$x - h = \pm (x + 5)$

Step 6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 6.1

First, use the positive value of the $\pm$ to find the first solution.

$x - h = x + 5$

Step 6.2

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

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Step 6.2.1

Subtract $x$ from both sides of the equation.

$- h = x + 5 - x$

Step 6.2.2

Combine the opposite terms in $x + 5 - x$ .

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Step 6.2.2.1

Subtract $x$ from $x$ .

$- h = 0 + 5$

Step 6.2.2.2

Add $0$ and $5$ .

$- h = 5$

Step 6.3

Divide each term in $- h = 5$ by $- 1$ and simplify.

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Step 6.3.1

Divide each term in $- h = 5$ by $- 1$ .

$\frac{- h}{- 1} = \frac{5}{- 1}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Dividing two negative values results in a positive value.

$\frac{h}{1} = \frac{5}{- 1}$

Step 6.3.2.2

Divide $h$ by $1$ .

$h = \frac{5}{- 1}$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Divide $5$ by $- 1$ .

$h = - 5$

Step 6.4

Next, use the negative value of the $\pm$ to find the second solution.

$x - h = - (x + 5)$

Step 6.5

Simplify $- (x + 5)$ .

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Step 6.5.1

Rewrite.

$x - h = 0 + 0 - (x + 5)$

Step 6.5.2

Simplify by adding zeros.

$x - h = - (x + 5)$

Step 6.5.3

Apply the distributive property.

$x - h = - x - 1 \cdot 5$

Step 6.5.4

Multiply $- 1$ by $5$ .

$x - h = - x - 5$

Step 6.6

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

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Step 6.6.1

Subtract $x$ from both sides of the equation.

$- h = - x - 5 - x$

Step 6.6.2

Subtract $x$ from $- x$ .

$- h = - 2 x - 5$

Step 6.7

Divide each term in $- h = - 2 x - 5$ by $- 1$ and simplify.

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Step 6.7.1

Divide each term in $- h = - 2 x - 5$ by $- 1$ .

$\frac{- h}{- 1} = \frac{- 2 x}{- 1} + \frac{- 5}{- 1}$

Step 6.7.2

Simplify the left side.

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Step 6.7.2.1

Dividing two negative values results in a positive value.

$\frac{h}{1} = \frac{- 2 x}{- 1} + \frac{- 5}{- 1}$

Step 6.7.2.2

Divide $h$ by $1$ .

$h = \frac{- 2 x}{- 1} + \frac{- 5}{- 1}$

Step 6.7.3

Simplify the right side.

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Step 6.7.3.1

Simplify each term.

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Step 6.7.3.1.1

Move the negative one from the denominator of $\frac{- 2 x}{- 1}$ .

$h = - 1 \cdot (- 2 x) + \frac{- 5}{- 1}$

Step 6.7.3.1.2

Rewrite $- 1 \cdot (- 2 x)$ as $- (- 2 x)$ .

$h = - (- 2 x) + \frac{- 5}{- 1}$

Step 6.7.3.1.3

Multiply $- 2$ by $- 1$ .

$h = 2 x + \frac{- 5}{- 1}$

Step 6.7.3.1.4

Divide $- 5$ by $- 1$ .

$h = 2 x + 5$

Step 6.8

The complete solution is the result of both the positive and negative portions of the solution.

$h = - 5$

$h = 2 x + 5$

$h = - 5$

$h = 2 x + 5$