Algebra Examples

$x^{2} + 98 = 8 x + 7$

Step 1

Simplify the equation into a proper form to complete the square.

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Step 1.1

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

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Step 1.1.1

Subtract $98$ from both sides of the equation.

$x^{2} = 8 x + 7 - 98$

Step 1.1.2

Subtract $98$ from $7$ .

$x^{2} = 8 x - 91$

Step 1.2

Subtract $8 x$ from both sides of the equation.

$x^{2} - 8 x = - 91$

Step 2

To create a trinomial square on the left side of the equation, find a value that is equal to the square of half of $b$ .

${(\frac{b}{2})}^{2} = {(- 4)}^{2}$

Step 3

Add the term to each side of the equation.

$x^{2} - 8 x + {(- 4)}^{2} = - 91 + {(- 4)}^{2}$

Step 4

Simplify the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

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

$x^{2} - 8 x + 16 = - 91 + {(- 4)}^{2}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- 91 + {(- 4)}^{2}$ .

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Step 4.2.1.1

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

$x^{2} - 8 x + 16 = - 91 + 16$

Step 4.2.1.2

Add $- 91$ and $16$ .

$x^{2} - 8 x + 16 = - 75$

Step 5

Factor the perfect trinomial square into ${(x - 4)}^{2}$ .

${(x - 4)}^{2} = - 75$

Step 6

Solve the equation for $x$ .

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Step 6.1

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

$x - 4 = \pm \sqrt{- 75}$

Step 6.2

Simplify $\pm \sqrt{- 75}$ .

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Step 6.2.1

Rewrite $- 75$ as $- 1 (75)$ .

$x - 4 = \pm \sqrt{- 1 (75)}$

Step 6.2.2

Rewrite $\sqrt{- 1 (75)}$ as $\sqrt{- 1} \cdot \sqrt{75}$ .

$x - 4 = \pm \sqrt{- 1} \cdot \sqrt{75}$

Step 6.2.3

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

$x - 4 = \pm i \cdot \sqrt{75}$

Step 6.2.4

Rewrite $75$ as $5^{2} \cdot 3$ .

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Step 6.2.4.1

Factor $25$ out of $75$ .

$x - 4 = \pm i \cdot \sqrt{25 (3)}$

Step 6.2.4.2

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

$x - 4 = \pm i \cdot \sqrt{5^{2} \cdot 3}$

Step 6.2.5

Pull terms out from under the radical.

$x - 4 = \pm i \cdot (5 \sqrt{3})$

Step 6.2.6

Move $5$ to the left of $i$ .

$x - 4 = \pm 5 i \sqrt{3}$

Step 6.3

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

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Step 6.3.1

Add $4$ to both sides of the equation.

$x = \pm 5 i \sqrt{3} + 4$

Step 6.3.2

Reorder $\pm 5 i \sqrt{3}$ and $4$ .

$x = 4 \pm 5 i \sqrt{3}$