Solve Using the Quadratic Formula x^2+13/2x+15/2=0

$x^{2} + \frac{13}{2} x + \frac{15}{2} = 0$

Step 1

Simplify the left side.

Combine $\frac{13}{2}$ and $x$ .

$x^{2} + \frac{13 x}{2} + \frac{15}{2} = 0$

Step 2

Multiply through by the least common denominator $2$ , then simplify.

Apply the distributive property.

$2 x^{2} + 2 (\frac{13 x}{2}) + 2 (\frac{15}{2}) = 0$

Simplify.

Cancel the common factor of $2$ .

Cancel the common factor.

$2 x^{2} + 2 (\frac{13 x}{2}) + 2 (\frac{15}{2}) = 0$

Rewrite the expression.

$2 x^{2} + 13 x + 2 (\frac{15}{2}) = 0$

Cancel the common factor of $2$ .

Cancel the common factor.

$2 x^{2} + 13 x + 2 (\frac{15}{2}) = 0$

Rewrite the expression.

$2 x^{2} + 13 x + 15 = 0$

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 $a = 2$ , $b = 13$ , and $c = 15$ into the quadratic formula and solve for $x$ .

$\frac{- 13 \pm \sqrt{13^{2} - 4 \cdot (2 \cdot 15)}}{2 \cdot 2}$

Step 5

Simplify.

Simplify the numerator.

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

$x = \frac{- 13 \pm \sqrt{169 - 4 \cdot 2 \cdot 15}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot 15$ .

Multiply $- 4$ by $2$ .

$x = \frac{- 13 \pm \sqrt{169 - 8 \cdot 15}}{2 \cdot 2}$

Multiply $- 8$ by $15$ .

$x = \frac{- 13 \pm \sqrt{169 - 120}}{2 \cdot 2}$

Subtract $120$ from $169$ .

$x = \frac{- 13 \pm \sqrt{49}}{2 \cdot 2}$

Rewrite $49$ as $7^{2}$ .

$x = \frac{- 13 \pm \sqrt{7^{2}}}{2 \cdot 2}$

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

$x = \frac{- 13 \pm 7}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{- 13 \pm 7}{4}$

Step 6

The final answer is the combination of both solutions.

$x = - \frac{3}{2}, - 5$

Solve Using the Quadratic Formula x^2+13/2x+15/2=0

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