Algebra Examples

$\frac{1}{7} y^{2} = - y + \frac{1}{14}$

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Simplify the left side.

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Step 1.1.1

Combine $\frac{1}{7}$ and $y^{2}$ .

$\frac{y^{2}}{7} = - y + \frac{1}{14}$

Step 1.2

Move all the expressions to the left side of the equation.

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Step 1.2.1

Add $y$ to both sides of the equation.

$\frac{y^{2}}{7} + y = \frac{1}{14}$

Step 1.2.2

Subtract $\frac{1}{14}$ from both sides of the equation.

$\frac{y^{2}}{7} + y - \frac{1}{14} = 0$

Step 2

Multiply through by the least common denominator $14$ , then simplify.

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Step 2.1

Apply the distributive property.

$14 (\frac{y^{2}}{7}) + 14 y + 14 (- \frac{1}{14}) = 0$

Step 2.2

Simplify.

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Step 2.2.1

Cancel the common factor of $7$ .

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Step 2.2.1.1

Factor $7$ out of $14$ .

$7 (2) (\frac{y^{2}}{7}) + 14 y + 14 (- \frac{1}{14}) = 0$

Step 2.2.1.2

Cancel the common factor.

$7 \cdot (2 (\frac{y^{2}}{7})) + 14 y + 14 (- \frac{1}{14}) = 0$

Step 2.2.1.3

Rewrite the expression.

$2 y^{2} + 14 y + 14 (- \frac{1}{14}) = 0$

Step 2.2.2

Cancel the common factor of $14$ .

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Step 2.2.2.1

Move the leading negative in $- \frac{1}{14}$ into the numerator.

$2 y^{2} + 14 y + 14 (\frac{- 1}{14}) = 0$

Step 2.2.2.2

Cancel the common factor.

$2 y^{2} + 14 y + 14 (\frac{- 1}{14}) = 0$

Step 2.2.2.3

Rewrite the expression.

$2 y^{2} + 14 y - 1 = 0$

Step 3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4

Substitute the values $a = 2$ , $b = 14$ , and $c = - 1$ into the quadratic formula and solve for $y$ .

$\frac{- 14 \pm \sqrt{14^{2} - 4 \cdot (2 \cdot - 1)}}{2 \cdot 2}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $14$ to the power of $2$ .

$y = \frac{- 14 \pm \sqrt{196 - 4 \cdot 2 \cdot - 1}}{2 \cdot 2}$

Step 5.1.2

Multiply $- 4 \cdot 2 \cdot - 1$ .

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Step 5.1.2.1

Multiply $- 4$ by $2$ .

$y = \frac{- 14 \pm \sqrt{196 - 8 \cdot - 1}}{2 \cdot 2}$

Step 5.1.2.2

Multiply $- 8$ by $- 1$ .

$y = \frac{- 14 \pm \sqrt{196 + 8}}{2 \cdot 2}$

Step 5.1.3

Add $196$ and $8$ .

$y = \frac{- 14 \pm \sqrt{204}}{2 \cdot 2}$

Step 5.1.4

Rewrite $204$ as $2^{2} \cdot 51$ .

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Step 5.1.4.1

Factor $4$ out of $204$ .

$y = \frac{- 14 \pm \sqrt{4 (51)}}{2 \cdot 2}$

Step 5.1.4.2

Rewrite $4$ as $2^{2}$ .

$y = \frac{- 14 \pm \sqrt{2^{2} \cdot 51}}{2 \cdot 2}$

Step 5.1.5

Pull terms out from under the radical.

$y = \frac{- 14 \pm 2 \sqrt{51}}{2 \cdot 2}$

Step 5.2

Multiply $2$ by $2$ .

$y = \frac{- 14 \pm 2 \sqrt{51}}{4}$

Step 5.3

Simplify $\frac{- 14 \pm 2 \sqrt{51}}{4}$ .

$y = \frac{- 7 \pm \sqrt{51}}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$y = \frac{- 7 \pm \sqrt{51}}{2}$

Decimal Form:

$y = 0.07071421 \dots, - 7.07071421 \dots$