Solve for x f(10)=1/2x^2+2x-5

$f (10) = \frac{1}{2} x^{2} + 2 x - 5$

Step 1

Rewrite the equation as $\frac{1}{2} x^{2} + 2 x - 5 = f (10)$ .

$\frac{1}{2} x^{2} + 2 x - 5 = f (10)$

Step 2

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

$\frac{x^{2}}{2} + 2 x - 5 = f (10)$

Step 3

Move $10$ to the left of $f$ .

$\frac{x^{2}}{2} + 2 x - 5 = 10 f$

Step 4

Subtract $10 f$ from both sides of the equation.

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

Step 5

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

Apply the distributive property.

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

Simplify.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$x^{2} + 2 (2 x) + 2 \cdot - 5 + 2 (- 10 f) = 0$

Multiply $2$ by $2$ .

$x^{2} + 4 x + 2 \cdot - 5 + 2 (- 10 f) = 0$

Multiply $2$ by $- 5$ .

$x^{2} + 4 x - 10 + 2 (- 10 f) = 0$

Multiply $- 10$ by $2$ .

$x^{2} + 4 x - 10 - 20 f = 0$

Move $- 10$ .

$x^{2} + 4 x - 20 f - 10 = 0$

Move $4 x$ .

$x^{2} - 20 f + 4 x - 10 = 0$

Step 6

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 7

Substitute the values $a = 1$ , $b = 4$ , and $c = - 20 f - 10$ into the quadratic formula and solve for $x$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (1 \cdot (- 20 f - 10))}}{2 \cdot 1}$

Step 8

Simplify.

Simplify the numerator.

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot (- 20 f - 10)}}{2 \cdot 1}$

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot (- 20 f - 10)}}{2 \cdot 1}$

Apply the distributive property.

$x = \frac{- 4 \pm \sqrt{16 - 4 (- 20 f) - 4 \cdot - 10}}{2 \cdot 1}$

Multiply $- 20$ by $- 4$ .

$x = \frac{- 4 \pm \sqrt{16 + 80 f - 4 \cdot - 10}}{2 \cdot 1}$

Multiply $- 4$ by $- 10$ .

$x = \frac{- 4 \pm \sqrt{16 + 80 f + 40}}{2 \cdot 1}$

Add $16$ and $40$ .

$x = \frac{- 4 \pm \sqrt{80 f + 56}}{2 \cdot 1}$

Factor $8$ out of $80 f + 56$ .

Factor $8$ out of $80 f$ .

$x = \frac{- 4 \pm \sqrt{8 (10 f) + 56}}{2 \cdot 1}$

Factor $8$ out of $56$ .

$x = \frac{- 4 \pm \sqrt{8 (10 f) + 8 (7)}}{2 \cdot 1}$

Factor $8$ out of $8 (10 f) + 8 (7)$ .

$x = \frac{- 4 \pm \sqrt{8 (10 f + 7)}}{2 \cdot 1}$

Rewrite $8 (10 f + 7)$ as $2^{2} \cdot (2 (10 f + 7))$ .

Factor $4$ out of $8$ .

$x = \frac{- 4 \pm \sqrt{4 (2) (10 f + 7)}}{2 \cdot 1}$

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

$x = \frac{- 4 \pm \sqrt{2^{2} \cdot (2 (10 f + 7))}}{2 \cdot 1}$

Add parentheses.

$x = \frac{- 4 \pm \sqrt{2^{2} \cdot (2 (10 f + 7))}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 4 \pm 2 \sqrt{2 (10 f + 7)}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 2 \sqrt{2 (10 f + 7)}}{2}$

Simplify $\frac{- 4 \pm 2 \sqrt{2 (10 f + 7)}}{2}$ .

$x = - 2 \pm \sqrt{2 (10 f + 7)}$

Step 9

The final answer is the combination of both solutions.

$x = - 2 + \sqrt{2 (10 f + 7)}$

$x = - 2 - \sqrt{2 (10 f + 7)}$

Solve for x f(10)=1/2x^2+2x-5

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