Solve Using the Quadratic Formula f(x)=1/2x^2+3x+7

$f (x) = \frac{1}{2} x^{2} + 3 x + 7$

Step 1

Simplify the left side.

Combine $\frac{1}{2}$ and $x^{2}$ .

$\frac{x^{2}}{2} + 3 x + 7 = 0$

Step 2

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

Apply the distributive property.

$2 (\frac{x^{2}}{2}) + 2 (3 x) + 2 \cdot 7 = 0$

Simplify.

Cancel the common factor of $2$ .

Cancel the common factor.

$2 (\frac{x^{2}}{2}) + 2 (3 x) + 2 \cdot 7 = 0$

Rewrite the expression.

$x^{2} + 2 (3 x) + 2 \cdot 7 = 0$

Multiply $3$ by $2$ .

$x^{2} + 6 x + 2 \cdot 7 = 0$

Multiply $2$ by $7$ .

$x^{2} + 6 x + 14 = 0$

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 $a = 1$ , $b = 6$ , and $c = 14$ into the quadratic formula and solve for $x$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (1 \cdot 14)}}{2 \cdot 1}$

Step 5

Simplify.

Simplify the numerator.

Raise $6$ to the power of $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1 \cdot 14}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 14$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 14}}{2 \cdot 1}$

Multiply $- 4$ by $14$ .

$x = \frac{- 6 \pm \sqrt{36 - 56}}{2 \cdot 1}$

Subtract $56$ from $36$ .

$x = \frac{- 6 \pm \sqrt{- 20}}{2 \cdot 1}$

Rewrite $- 20$ as $- 1 (20)$ .

$x = \frac{- 6 \pm \sqrt{- 1 \cdot 20}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (20)}$ as $\sqrt{- 1} \cdot \sqrt{20}$ .

$x = \frac{- 6 \pm \sqrt{- 1} \cdot \sqrt{20}}{2 \cdot 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 6 \pm i \cdot \sqrt{20}}{2 \cdot 1}$

Rewrite $20$ as $2^{2} \cdot 5$ .

Factor $4$ out of $20$ .

$x = \frac{- 6 \pm i \cdot \sqrt{4 (5)}}{2 \cdot 1}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 6 \pm i \cdot \sqrt{2^{2} \cdot 5}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 6 \pm i \cdot (2 \sqrt{5})}{2 \cdot 1}$

Move $2$ to the left of $i$ .

$x = \frac{- 6 \pm 2 i \sqrt{5}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 6 \pm 2 i \sqrt{5}}{2}$

Simplify $\frac{- 6 \pm 2 i \sqrt{5}}{2}$ .

$x = - 3 \pm i \sqrt{5}$

Solve Using the Quadratic Formula f(x)=1/2x^2+3x+7

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