Precalculus Examples

$y^{2} + 8 y - 2 = 0$

Step 1

Add $2$ to both sides of the equation.

$y^{2} + 8 y = 2$

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.

$y^{2} + 8 y + {(4)}^{2} = 2 + {(4)}^{2}$

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 $2$ .

$y^{2} + 8 y + 16 = 2 + {(4)}^{2}$

Step 4.2

Simplify the right side.

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Step 4.2.1

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

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Step 4.2.1.1

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

$y^{2} + 8 y + 16 = 2 + 16$

Step 4.2.1.2

Add $2$ and $16$ .

$y^{2} + 8 y + 16 = 18$

Step 5

Factor the perfect trinomial square into ${(y + 4)}^{2}$ .

${(y + 4)}^{2} = 18$

Step 6

Solve the equation for $y$ .

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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.

$y + 4 = \pm \sqrt{18}$

Step 6.2

Simplify $\pm \sqrt{18}$ .

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Step 6.2.1

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 6.2.1.1

Factor $9$ out of $18$ .

$y + 4 = \pm \sqrt{9 (2)}$

Step 6.2.1.2

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

$y + 4 = \pm \sqrt{3^{2} \cdot 2}$

Step 6.2.2

Pull terms out from under the radical.

$y + 4 = \pm 3 \sqrt{2}$

Step 6.3

Subtract $4$ from both sides of the equation.

$y = \pm 3 \sqrt{2} - 4$

Step 7

The result can be shown in multiple forms.

Exact Form:

$y = \pm 3 \sqrt{2} - 4$

Decimal Form:

$y = 0.24264068 \dots, - 8.24264068 \dots$