Solve Using the Quadratic Formula (x^2)/2+(13x)/20=3/2

$\frac{x^{2}}{2} + \frac{13 x}{20} = \frac{3}{2}$

Step 1

Subtract $\frac{3}{2}$ from both sides of the equation.

$\frac{x^{2}}{2} + \frac{13 x}{20} - \frac{3}{2} = 0$

Step 2

Multiply through by the least common denominator $20$ , then simplify.

Apply the distributive property.

$20 (\frac{x^{2}}{2}) + 20 (\frac{13 x}{20}) + 20 (- \frac{3}{2}) = 0$

Simplify.

Cancel the common factor of $2$ .

Factor $2$ out of $20$ .

$2 (10) (\frac{x^{2}}{2}) + 20 (\frac{13 x}{20}) + 20 (- \frac{3}{2}) = 0$

Cancel the common factor.

$2 \cdot (10 (\frac{x^{2}}{2})) + 20 (\frac{13 x}{20}) + 20 (- \frac{3}{2}) = 0$

Rewrite the expression.

$10 x^{2} + 20 (\frac{13 x}{20}) + 20 (- \frac{3}{2}) = 0$

Cancel the common factor of $20$ .

Cancel the common factor.

$10 x^{2} + 20 (\frac{13 x}{20}) + 20 (- \frac{3}{2}) = 0$

Rewrite the expression.

$10 x^{2} + 13 x + 20 (- \frac{3}{2}) = 0$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$10 x^{2} + 13 x + 20 (\frac{- 3}{2}) = 0$

Factor $2$ out of $20$ .

$10 x^{2} + 13 x + 2 (10) (\frac{- 3}{2}) = 0$

Cancel the common factor.

$10 x^{2} + 13 x + 2 \cdot (10 (\frac{- 3}{2})) = 0$

Rewrite the expression.

$10 x^{2} + 13 x + 10 \cdot - 3 = 0$

Multiply $10$ by $- 3$ .

$10 x^{2} + 13 x - 30 = 0$

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 $a = 10$ , $b = 13$ , and $c = - 30$ into the quadratic formula and solve for $x$ .

$\frac{- 13 \pm \sqrt{13^{2} - 4 \cdot (10 \cdot - 30)}}{2 \cdot 10}$

Step 5

Simplify.

Simplify the numerator.

Raise $13$ to the power of $2$ .

$x = \frac{- 13 \pm \sqrt{169 - 4 \cdot 10 \cdot - 30}}{2 \cdot 10}$

Multiply $- 4 \cdot 10 \cdot - 30$ .

Multiply $- 4$ by $10$ .

$x = \frac{- 13 \pm \sqrt{169 - 40 \cdot - 30}}{2 \cdot 10}$

Multiply $- 40$ by $- 30$ .

$x = \frac{- 13 \pm \sqrt{169 + 1200}}{2 \cdot 10}$

Add $169$ and $1200$ .

$x = \frac{- 13 \pm \sqrt{1369}}{2 \cdot 10}$

Rewrite $1369$ as $37^{2}$ .

$x = \frac{- 13 \pm \sqrt{37^{2}}}{2 \cdot 10}$

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

$x = \frac{- 13 \pm 37}{2 \cdot 10}$

Multiply $2$ by $10$ .

$x = \frac{- 13 \pm 37}{20}$

Step 6

The final answer is the combination of both solutions.

$x = \frac{6}{5}, - \frac{5}{2}$

Solve Using the Quadratic Formula (x^2)/2+(13x)/20=3/2

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