Algebra Examples

$\frac{3}{7} y^{2} + \frac{5}{7} y = \frac{8}{7}$

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

Simplify each term.

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Step 1.1.1.1

Combine $\frac{3}{7}$ and $y^{2}$ .

$\frac{3 y^{2}}{7} + \frac{5}{7} y = \frac{8}{7}$

Step 1.1.1.2

Combine $\frac{5}{7}$ and $y$ .

$\frac{3 y^{2}}{7} + \frac{5 y}{7} = \frac{8}{7}$

Step 1.2

Subtract $\frac{8}{7}$ from both sides of the equation.

$\frac{3 y^{2}}{7} + \frac{5 y}{7} - \frac{8}{7} = 0$

Step 2

Multiply through by the least common denominator $7$ , then simplify.

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Step 2.1

Apply the distributive property.

$7 (\frac{3 y^{2}}{7}) + 7 (\frac{5 y}{7}) + 7 (- \frac{8}{7}) = 0$

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

Cancel the common factor.

$7 (\frac{3 y^{2}}{7}) + 7 (\frac{5 y}{7}) + 7 (- \frac{8}{7}) = 0$

Step 2.2.1.2

Rewrite the expression.

$3 y^{2} + 7 (\frac{5 y}{7}) + 7 (- \frac{8}{7}) = 0$

Step 2.2.2

Cancel the common factor of $7$ .

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Step 2.2.2.1

Cancel the common factor.

$3 y^{2} + 7 (\frac{5 y}{7}) + 7 (- \frac{8}{7}) = 0$

Step 2.2.2.2

Rewrite the expression.

$3 y^{2} + 5 y + 7 (- \frac{8}{7}) = 0$

Step 2.2.3

Cancel the common factor of $7$ .

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Step 2.2.3.1

Move the leading negative in $- \frac{8}{7}$ into the numerator.

$3 y^{2} + 5 y + 7 (\frac{- 8}{7}) = 0$

Step 2.2.3.2

Cancel the common factor.

$3 y^{2} + 5 y + 7 (\frac{- 8}{7}) = 0$

Step 2.2.3.3

Rewrite the expression.

$3 y^{2} + 5 y - 8 = 0$

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 $a = 3$ , $b = 5$ , and $c = - 8$ into the quadratic formula and solve for $y$ .

$\frac{- 5 \pm \sqrt{5^{2} - 4 \cdot (3 \cdot - 8)}}{2 \cdot 3}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

$y = \frac{- 5 \pm \sqrt{25 - 4 \cdot 3 \cdot - 8}}{2 \cdot 3}$

Step 5.1.2

Multiply $- 4 \cdot 3 \cdot - 8$ .

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Step 5.1.2.1

Multiply $- 4$ by $3$ .

$y = \frac{- 5 \pm \sqrt{25 - 12 \cdot - 8}}{2 \cdot 3}$

Step 5.1.2.2

Multiply $- 12$ by $- 8$ .

$y = \frac{- 5 \pm \sqrt{25 + 96}}{2 \cdot 3}$

Step 5.1.3

Add $25$ and $96$ .

$y = \frac{- 5 \pm \sqrt{121}}{2 \cdot 3}$

Step 5.1.4

Rewrite $121$ as $11^{2}$ .

$y = \frac{- 5 \pm \sqrt{11^{2}}}{2 \cdot 3}$

Step 5.1.5

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

$y = \frac{- 5 \pm 11}{2 \cdot 3}$

Step 5.2

Multiply $2$ by $3$ .

$y = \frac{- 5 \pm 11}{6}$

Step 6

The final answer is the combination of both solutions.

$y = 1, - \frac{8}{3}$