Solve -4x^2+12=-6x-8

$$- 4 x ^ { 2 } + 12 = - 6 x - 8$$

Solve for x

$$x = \frac{\sqrt{89} + 3}{4} \approx 3.108495283$$

$$x=\frac{3-\sqrt{89}}{4}\approx -1.608495283$$

Steps Using the Quadratic Formula

Add 6x to both sides.

$$-4x^{2}+12+6x=-8$$

Add 8 to both sides.

$$-4x^{2}+12+6x+8=0$$

Add 12 and 8 to get 20.

$$-4x^{2}+20+6x=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-4x^{2}+6x+20=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 6 for b, and 20 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\left(-4\right)\times 20}}{2\left(-4\right)}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\left(-4\right)\times 20}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-6±\sqrt{36+16\times 20}}{2\left(-4\right)}$$

Multiply 16 times 20.

$$x=\frac{-6±\sqrt{36+320}}{2\left(-4\right)}$$

Add 36 to 320.

$$x=\frac{-6±\sqrt{356}}{2\left(-4\right)}$$

Take the square root of 356.

$$x=\frac{-6±2\sqrt{89}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-6±2\sqrt{89}}{-8}$$

Now solve the equation x=\frac{-6±2\sqrt{89}}{-8} when ± is plus. Add -6 to 2\sqrt{89}.

$$x=\frac{2\sqrt{89}-6}{-8}$$

Divide -6+2\sqrt{89} by -8.

$$x=\frac{3-\sqrt{89}}{4}$$

Now solve the equation x=\frac{-6±2\sqrt{89}}{-8} when ± is minus. Subtract 2\sqrt{89} from -6.

$$x=\frac{-2\sqrt{89}-6}{-8}$$

Divide -6-2\sqrt{89} by -8.

$$x=\frac{\sqrt{89}+3}{4}$$

The equation is now solved.

$$x=\frac{3-\sqrt{89}}{4} x=\frac{\sqrt{89}+3}{4}$$

Steps for Completing the Square

Add 6x to both sides.

$$-4x^{2}+12+6x=-8$$

Subtract 12 from both sides.

$$-4x^{2}+6x=-8-12$$

Subtract 12 from -8 to get -20.

$$-4x^{2}+6x=-20$$

Divide both sides by -4.

$$\frac{-4x^{2}+6x}{-4}=-\frac{20}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\frac{6}{-4}x=-\frac{20}{-4}$$

Reduce the fraction \frac{6}{-4} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{3}{2}x=-\frac{20}{-4}$$

Divide -20 by -4.

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

Divide -\frac{3}{2}, the coefficient of the x term, by 2 to get -\frac{3}{4}. Then add the square of -\frac{3}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{3}{2}x+\left(-\frac{3}{4}\right)^{2}=5+\left(-\frac{3}{4}\right)^{2}$$

Square -\frac{3}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=5+\frac{9}{16}$$

Add 5 to \frac{9}{16}.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{89}{16}$$

Factor x^{2}-\frac{3}{2}x+\frac{9}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{3}{4}\right)^{2}=\frac{89}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{4}\right)^{2}}=\sqrt{\frac{89}{16}}$$

Simplify.

$$x-\frac{3}{4}=\frac{\sqrt{89}}{4} x-\frac{3}{4}=-\frac{\sqrt{89}}{4}$$

Add \frac{3}{4} to both sides of the equation.

$$x=\frac{\sqrt{89}+3}{4} x=\frac{3-\sqrt{89}}{4}$$