Solve Using the Quadratic Formula 2/5y^2+7/5y=9/5

$\frac{2}{5} y^{2} + \frac{7}{5} y = \frac{9}{5}$

Step 1

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

Simplify the left side.

Simplify each term.

Combine $\frac{2}{5}$ and $y^{2}$ .

$\frac{2 y^{2}}{5} + \frac{7}{5} y = \frac{9}{5}$

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

$\frac{2 y^{2}}{5} + \frac{7 y}{5} = \frac{9}{5}$

Subtract $\frac{9}{5}$ from both sides of the equation.

$\frac{2 y^{2}}{5} + \frac{7 y}{5} - \frac{9}{5} = 0$

Step 2

Multiply through by the least common denominator $5$ , then simplify.

Apply the distributive property.

$5 (\frac{2 y^{2}}{5}) + 5 (\frac{7 y}{5}) + 5 (- \frac{9}{5}) = 0$

Simplify.

Cancel the common factor of $5$ .

Cancel the common factor.

$5 (\frac{2 y^{2}}{5}) + 5 (\frac{7 y}{5}) + 5 (- \frac{9}{5}) = 0$

Rewrite the expression.

$2 y^{2} + 5 (\frac{7 y}{5}) + 5 (- \frac{9}{5}) = 0$

Cancel the common factor of $5$ .

Cancel the common factor.

$2 y^{2} + 5 (\frac{7 y}{5}) + 5 (- \frac{9}{5}) = 0$

Rewrite the expression.

$2 y^{2} + 7 y + 5 (- \frac{9}{5}) = 0$

Cancel the common factor of $5$ .

Move the leading negative in $- \frac{9}{5}$ into the numerator.

$2 y^{2} + 7 y + 5 (\frac{- 9}{5}) = 0$

Cancel the common factor.

$2 y^{2} + 7 y + 5 (\frac{- 9}{5}) = 0$

Rewrite the expression.

$2 y^{2} + 7 y - 9 = 0$

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 $a = 2$ , $b = 7$ , and $c = - 9$ into the quadratic formula and solve for $y$ .

$\frac{- 7 \pm \sqrt{7^{2} - 4 \cdot (2 \cdot - 9)}}{2 \cdot 2}$

Step 5

Simplify.

Simplify the numerator.

Raise $7$ to the power of $2$ .

$y = \frac{- 7 \pm \sqrt{49 - 4 \cdot 2 \cdot - 9}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot - 9$ .

Multiply $- 4$ by $2$ .

$y = \frac{- 7 \pm \sqrt{49 - 8 \cdot - 9}}{2 \cdot 2}$

Multiply $- 8$ by $- 9$ .

$y = \frac{- 7 \pm \sqrt{49 + 72}}{2 \cdot 2}$

Add $49$ and $72$ .

$y = \frac{- 7 \pm \sqrt{121}}{2 \cdot 2}$

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

$y = \frac{- 7 \pm \sqrt{11^{2}}}{2 \cdot 2}$

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

$y = \frac{- 7 \pm 11}{2 \cdot 2}$

Multiply $2$ by $2$ .

$y = \frac{- 7 \pm 11}{4}$

Step 6

The final answer is the combination of both solutions.

$y = 1, - \frac{9}{2}$

Solve Using the Quadratic Formula 2/5y^2+7/5y=9/5

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