Solve Using the Quadratic Formula 1/10y^2=y-5/2

$\frac{1}{10} y^{2} = y - \frac{5}{2}$

Step 1

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

Simplify the left side.

Combine $\frac{1}{10}$ and $y^{2}$ .

$\frac{y^{2}}{10} = y - \frac{5}{2}$

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

Subtract $y$ from both sides of the equation.

$\frac{y^{2}}{10} - y = - \frac{5}{2}$

Add $\frac{5}{2}$ to both sides of the equation.

$\frac{y^{2}}{10} - y + \frac{5}{2} = 0$

Step 2

Multiply through by the least common denominator $10$ , then simplify.

Apply the distributive property.

$10 (\frac{y^{2}}{10}) + 10 (- y) + 10 (\frac{5}{2}) = 0$

Simplify.

Cancel the common factor of $10$ .

Cancel the common factor.

$10 (\frac{y^{2}}{10}) + 10 (- y) + 10 (\frac{5}{2}) = 0$

Rewrite the expression.

$y^{2} + 10 (- y) + 10 (\frac{5}{2}) = 0$

Multiply $- 1$ by $10$ .

$y^{2} - 10 y + 10 (\frac{5}{2}) = 0$

Cancel the common factor of $2$ .

Factor $2$ out of $10$ .

$y^{2} - 10 y + 2 (5) (\frac{5}{2}) = 0$

Cancel the common factor.

$y^{2} - 10 y + 2 \cdot (5 (\frac{5}{2})) = 0$

Rewrite the expression.

$y^{2} - 10 y + 5 \cdot 5 = 0$

Multiply $5$ by $5$ .

$y^{2} - 10 y + 25 = 0$

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 $a = 1$ , $b = - 10$ , and $c = 25$ into the quadratic formula and solve for $y$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (1 \cdot 25)}}{2 \cdot 1}$

Step 5

Simplify.

Simplify the numerator.

Raise $- 10$ to the power of $2$ .

$y = \frac{10 \pm \sqrt{100 - 4 \cdot 1 \cdot 25}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 25$ .

Multiply $- 4$ by $1$ .

$y = \frac{10 \pm \sqrt{100 - 4 \cdot 25}}{2 \cdot 1}$

Multiply $- 4$ by $25$ .

$y = \frac{10 \pm \sqrt{100 - 100}}{2 \cdot 1}$

Subtract $100$ from $100$ .

$y = \frac{10 \pm \sqrt{0}}{2 \cdot 1}$

Rewrite $0$ as $0^{2}$ .

$y = \frac{10 \pm \sqrt{0^{2}}}{2 \cdot 1}$

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

$y = \frac{10 \pm 0}{2 \cdot 1}$

$10$ plus or minus $0$ is $10$ .

$y = \frac{10}{2 \cdot 1}$

Multiply $2$ by $1$ .

$y = \frac{10}{2}$

Divide $10$ by $2$ .

$y = 5$

Step 6

The final answer is the combination of both solutions.

$y = 5$ Double roots

Solve Using the Quadratic Formula 1/10y^2=y-5/2

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