Solve Using the Quadratic Formula 3/7y^2+2/7y=16/7

$\frac{3}{7} y^{2} + \frac{2}{7} y = \frac{16}{7}$

Step 1

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

Simplify the left side.

Simplify each term.

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

$\frac{3 y^{2}}{7} + \frac{2}{7} y = \frac{16}{7}$

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

$\frac{3 y^{2}}{7} + \frac{2 y}{7} = \frac{16}{7}$

Subtract $\frac{16}{7}$ from both sides of the equation.

$\frac{3 y^{2}}{7} + \frac{2 y}{7} - \frac{16}{7} = 0$

Step 2

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

Apply the distributive property.

$7 (\frac{3 y^{2}}{7}) + 7 (\frac{2 y}{7}) + 7 (- \frac{16}{7}) = 0$

Simplify.

Cancel the common factor of $7$ .

Cancel the common factor.

$7 (\frac{3 y^{2}}{7}) + 7 (\frac{2 y}{7}) + 7 (- \frac{16}{7}) = 0$

Rewrite the expression.

$3 y^{2} + 7 (\frac{2 y}{7}) + 7 (- \frac{16}{7}) = 0$

Cancel the common factor of $7$ .

Cancel the common factor.

$3 y^{2} + 7 (\frac{2 y}{7}) + 7 (- \frac{16}{7}) = 0$

Rewrite the expression.

$3 y^{2} + 2 y + 7 (- \frac{16}{7}) = 0$

Cancel the common factor of $7$ .

Move the leading negative in $- \frac{16}{7}$ into the numerator.

$3 y^{2} + 2 y + 7 (\frac{- 16}{7}) = 0$

Cancel the common factor.

$3 y^{2} + 2 y + 7 (\frac{- 16}{7}) = 0$

Rewrite the expression.

$3 y^{2} + 2 y - 16 = 0$

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 $a = 3$ , $b = 2$ , and $c = - 16$ into the quadratic formula and solve for $y$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (3 \cdot - 16)}}{2 \cdot 3}$

Step 5

Simplify.

Simplify the numerator.

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

$y = \frac{- 2 \pm \sqrt{4 - 4 \cdot 3 \cdot - 16}}{2 \cdot 3}$

Multiply $- 4 \cdot 3 \cdot - 16$ .

Multiply $- 4$ by $3$ .

$y = \frac{- 2 \pm \sqrt{4 - 12 \cdot - 16}}{2 \cdot 3}$

Multiply $- 12$ by $- 16$ .

$y = \frac{- 2 \pm \sqrt{4 + 192}}{2 \cdot 3}$

Add $4$ and $192$ .

$y = \frac{- 2 \pm \sqrt{196}}{2 \cdot 3}$

Rewrite $196$ as $14^{2}$ .

$y = \frac{- 2 \pm \sqrt{14^{2}}}{2 \cdot 3}$

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

$y = \frac{- 2 \pm 14}{2 \cdot 3}$

Multiply $2$ by $3$ .

$y = \frac{- 2 \pm 14}{6}$

Simplify $\frac{- 2 \pm 14}{6}$ .

$y = \frac{- 1 \pm 7}{3}$

Step 6

The final answer is the combination of both solutions.

$y = 2, - \frac{8}{3}$

Solve Using the Quadratic Formula 3/7y^2+2/7y=16/7

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