Finite Math Examples

$x^{2} - 12 x + 36 = 90$

Step 1

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

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Step 1.1

Subtract $36$ from both sides of the equation.

$x^{2} - 12 x = 90 - 36$

Step 1.2

Subtract $36$ from $90$ .

$x^{2} - 12 x = 54$

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 $b$ .

${(\frac{b}{2})}^{2} = {(- 6)}^{2}$

Step 3

Add the term to each side of the equation.

$x^{2} - 12 x + {(- 6)}^{2} = 54 + {(- 6)}^{2}$

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 $- 6$ to the power of $2$ .

$x^{2} - 12 x + 36 = 54 + {(- 6)}^{2}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $54 + {(- 6)}^{2}$ .

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Step 4.2.1.1

Raise $- 6$ to the power of $2$ .

$x^{2} - 12 x + 36 = 54 + 36$

Step 4.2.1.2

Add $54$ and $36$ .

$x^{2} - 12 x + 36 = 90$

Step 5

Factor the perfect trinomial square into ${(x - 6)}^{2}$ .

${(x - 6)}^{2} = 90$

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.

$x - 6 = \pm \sqrt{90}$

Step 6.2

Simplify $\pm \sqrt{90}$ .

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Step 6.2.1

Rewrite $90$ as $3^{2} \cdot 10$ .

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Step 6.2.1.1

Factor $9$ out of $90$ .

$x - 6 = \pm \sqrt{9 (10)}$

Step 6.2.1.2

Rewrite $9$ as $3^{2}$ .

$x - 6 = \pm \sqrt{3^{2} \cdot 10}$

Step 6.2.2

Pull terms out from under the radical.

$x - 6 = \pm 3 \sqrt{10}$

Step 6.3

Add $6$ to both sides of the equation.

$x = \pm 3 \sqrt{10} + 6$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \pm 3 \sqrt{10} + 6$

Decimal Form:

$x = 15.48683298 \dots, - 3.48683298 \dots$