Solve for x -F(x)=3x^2+2x-4

$- F (x) = 3 x^{2} + 2 x - 4$

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$3 x^{2} + 2 x - 4 = - F x$

Add $F x$ to both sides of the equation.

$3 x^{2} + 2 x - 4 + F x = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 3$ , $b = 2 + F$ , and $c = - 4$ into the quadratic formula and solve for $x$ .

$\frac{- (2 + F) \pm \sqrt{{(2 + F)}^{2} - 4 \cdot (3 \cdot - 4)}}{2 \cdot 3}$

Simplify.

Simplify the numerator.

Apply the distributive property.

$x = \frac{- 1 \cdot 2 - F \pm \sqrt{{(2 + F)}^{2} - 4 \cdot 3 \cdot - 4}}{2 \cdot 3}$

Multiply $- 1$ by $2$ .

$x = \frac{- 2 - F \pm \sqrt{{(2 + F)}^{2} - 4 \cdot 3 \cdot - 4}}{2 \cdot 3}$

Rewrite ${(2 + F)}^{2}$ as $(2 + F) (2 + F)$ .

$x = \frac{- 2 - F \pm \sqrt{(2 + F) (2 + F) - 4 \cdot 3 \cdot - 4}}{2 \cdot 3}$

Expand $(2 + F) (2 + F)$ using the FOIL Method.

Apply the distributive property.

$x = \frac{- 2 - F \pm \sqrt{2 (2 + F) + F (2 + F) - 4 \cdot 3 \cdot - 4}}{2 \cdot 3}$

Apply the distributive property.

$x = \frac{- 2 - F \pm \sqrt{2 \cdot 2 + 2 F + F (2 + F) - 4 \cdot 3 \cdot - 4}}{2 \cdot 3}$

Apply the distributive property.

$x = \frac{- 2 - F \pm \sqrt{2 \cdot 2 + 2 F + F \cdot 2 + F \cdot F - 4 \cdot 3 \cdot - 4}}{2 \cdot 3}$

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $2$ .

$x = \frac{- 2 - F \pm \sqrt{4 + 2 F + F \cdot 2 + F \cdot F - 4 \cdot 3 \cdot - 4}}{2 \cdot 3}$

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

$x = \frac{- 2 - F \pm \sqrt{4 + 2 F + 2 \cdot F + F \cdot F - 4 \cdot 3 \cdot - 4}}{2 \cdot 3}$

Multiply $F$ by $F$ .

$x = \frac{- 2 - F \pm \sqrt{4 + 2 F + 2 F + F^{2} - 4 \cdot 3 \cdot - 4}}{2 \cdot 3}$

Add $2 F$ and $2 F$ .

$x = \frac{- 2 - F \pm \sqrt{4 + 4 F + F^{2} - 4 \cdot 3 \cdot - 4}}{2 \cdot 3}$

Multiply $- 4$ by $3$ .

$x = \frac{- 2 - F \pm \sqrt{4 + 4 F + F^{2} - 12 \cdot - 4}}{2 \cdot 3}$

Multiply $- 12$ by $- 4$ .

$x = \frac{- 2 - F \pm \sqrt{4 + 4 F + F^{2} + 48}}{2 \cdot 3}$

Add $4$ and $48$ .

$x = \frac{- 2 - F \pm \sqrt{4 F + F^{2} + 52}}{2 \cdot 3}$

Reorder terms.

$x = \frac{- 2 - F \pm \sqrt{F^{2} + 4 F + 52}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{- 2 - F \pm \sqrt{F^{2} + 4 F + 52}}{6}$

The final answer is the combination of both solutions.

$x = - \frac{2 + F - \sqrt{F^{2} + 4 F + 52}}{6}$

$x = - \frac{2 + F + \sqrt{F^{2} + 4 F + 52}}{6}$

Solve for x -F(x)=3x^2+2x-4

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