Algebra Examples

$2 y^{2} + 10 y = - 9$

Step 1

Divide each term in $2 y^{2} + 10 y = - 9$ by $2$ and simplify.

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Step 1.1

Divide each term in $2 y^{2} + 10 y = - 9$ by $2$ .

$\frac{2 y^{2}}{2} + \frac{10 y}{2} = \frac{- 9}{2}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Cancel the common factor of $2$ .

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Step 1.2.1.1.1

Cancel the common factor.

$\frac{2 y^{2}}{2} + \frac{10 y}{2} = \frac{- 9}{2}$

Step 1.2.1.1.2

Divide $y^{2}$ by $1$ .

$y^{2} + \frac{10 y}{2} = \frac{- 9}{2}$

Step 1.2.1.2

Cancel the common factor of $10$ and $2$ .

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Step 1.2.1.2.1

Factor $2$ out of $10 y$ .

$y^{2} + \frac{2 (5 y)}{2} = \frac{- 9}{2}$

Step 1.2.1.2.2

Cancel the common factors.

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Step 1.2.1.2.2.1

Factor $2$ out of $2$ .

$y^{2} + \frac{2 (5 y)}{2 (1)} = \frac{- 9}{2}$

Step 1.2.1.2.2.2

Cancel the common factor.

$y^{2} + \frac{2 (5 y)}{2 \cdot 1} = \frac{- 9}{2}$

Step 1.2.1.2.2.3

Rewrite the expression.

$y^{2} + \frac{5 y}{1} = \frac{- 9}{2}$

Step 1.2.1.2.2.4

Divide $5 y$ by $1$ .

$y^{2} + 5 y = \frac{- 9}{2}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Move the negative in front of the fraction.

$y^{2} + 5 y = - \frac{9}{2}$

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 $b$ .

${(\frac{b}{2})}^{2} = {(\frac{5}{2})}^{2}$

Step 3

Add the term to each side of the equation.

$y^{2} + 5 y + {(\frac{5}{2})}^{2} = - \frac{9}{2} + {(\frac{5}{2})}^{2}$

Step 4

Simplify the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify each term.

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Step 4.1.1.1

Apply the product rule to $\frac{5}{2}$ .

$y^{2} + 5 y + \frac{5^{2}}{2^{2}} = - \frac{9}{2} + {(\frac{5}{2})}^{2}$

Step 4.1.1.2

Raise $5$ to the power of $2$ .

$y^{2} + 5 y + \frac{25}{2^{2}} = - \frac{9}{2} + {(\frac{5}{2})}^{2}$

Step 4.1.1.3

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

$y^{2} + 5 y + \frac{25}{4} = - \frac{9}{2} + {(\frac{5}{2})}^{2}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{9}{2} + {(\frac{5}{2})}^{2}$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Apply the product rule to $\frac{5}{2}$ .

$y^{2} + 5 y + \frac{25}{4} = - \frac{9}{2} + \frac{5^{2}}{2^{2}}$

Step 4.2.1.1.2

Raise $5$ to the power of $2$ .

$y^{2} + 5 y + \frac{25}{4} = - \frac{9}{2} + \frac{25}{2^{2}}$

Step 4.2.1.1.3

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

$y^{2} + 5 y + \frac{25}{4} = - \frac{9}{2} + \frac{25}{4}$

Step 4.2.1.2

To write $- \frac{9}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y^{2} + 5 y + \frac{25}{4} = - \frac{9}{2} \cdot \frac{2}{2} + \frac{25}{4}$

Step 4.2.1.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.1.3.1

Multiply $\frac{9}{2}$ by $\frac{2}{2}$ .

$y^{2} + 5 y + \frac{25}{4} = - \frac{9 \cdot 2}{2 \cdot 2} + \frac{25}{4}$

Step 4.2.1.3.2

Multiply $2$ by $2$ .

$y^{2} + 5 y + \frac{25}{4} = - \frac{9 \cdot 2}{4} + \frac{25}{4}$

Step 4.2.1.4

Combine the numerators over the common denominator.

$y^{2} + 5 y + \frac{25}{4} = \frac{- 9 \cdot 2 + 25}{4}$

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply $- 9$ by $2$ .

$y^{2} + 5 y + \frac{25}{4} = \frac{- 18 + 25}{4}$

Step 4.2.1.5.2

Add $- 18$ and $25$ .

$y^{2} + 5 y + \frac{25}{4} = \frac{7}{4}$

Step 5

Factor the perfect trinomial square into ${(y + \frac{5}{2})}^{2}$ .

${(y + \frac{5}{2})}^{2} = \frac{7}{4}$

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.

$y + \frac{5}{2} = \pm \sqrt{\frac{7}{4}}$

Step 6.2

Simplify $\pm \sqrt{\frac{7}{4}}$ .

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Step 6.2.1

Rewrite $\sqrt{\frac{7}{4}}$ as $\frac{\sqrt{7}}{\sqrt{4}}$ .

$y + \frac{5}{2} = \pm \frac{\sqrt{7}}{\sqrt{4}}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

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

$y + \frac{5}{2} = \pm \frac{\sqrt{7}}{\sqrt{2^{2}}}$

Step 6.2.2.2

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

$y + \frac{5}{2} = \pm \frac{\sqrt{7}}{2}$

Step 6.3

Subtract $\frac{5}{2}$ from both sides of the equation.

$y = \pm \frac{\sqrt{7}}{2} - \frac{5}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$y = \pm \frac{\sqrt{7}}{2} - \frac{5}{2}$

Decimal Form:

$y = - 1.17712434 \dots, - 3.82287565 \dots$