Solve -2x^2+6x=x^2-2x-24

$$- 2 x ^ { 2 } + 6 x = x ^ { 2 } - 2 x - 24$$

Solve for x

$$x = \frac{2 \sqrt{22} + 4}{3} \approx 4.460277173$$

$$x=\frac{4-2\sqrt{22}}{3}\approx -1.793610507$$

Steps Using the Quadratic Formula

Subtract x^{2} from both sides.

$$-2x^{2}+6x-x^{2}=-2x-24$$

Combine -2x^{2} and -x^{2} to get -3x^{2}.

$$-3x^{2}+6x=-2x-24$$

Add 2x to both sides.

$$-3x^{2}+6x+2x=-24$$

Combine 6x and 2x to get 8x.

$$-3x^{2}+8x=-24$$

Add 24 to both sides.

$$-3x^{2}+8x+24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 8 for b, and 24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\left(-3\right)\times 24}}{2\left(-3\right)}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\left(-3\right)\times 24}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-8±\sqrt{64+12\times 24}}{2\left(-3\right)}$$

Multiply 12 times 24.

$$x=\frac{-8±\sqrt{64+288}}{2\left(-3\right)}$$

Add 64 to 288.

$$x=\frac{-8±\sqrt{352}}{2\left(-3\right)}$$

Take the square root of 352.

$$x=\frac{-8±4\sqrt{22}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-8±4\sqrt{22}}{-6}$$

Now solve the equation x=\frac{-8±4\sqrt{22}}{-6} when ± is plus. Add -8 to 4\sqrt{22}.

$$x=\frac{4\sqrt{22}-8}{-6}$$

Divide -8+4\sqrt{22} by -6.

$$x=\frac{4-2\sqrt{22}}{3}$$

Now solve the equation x=\frac{-8±4\sqrt{22}}{-6} when ± is minus. Subtract 4\sqrt{22} from -8.

$$x=\frac{-4\sqrt{22}-8}{-6}$$

Divide -8-4\sqrt{22} by -6.

$$x=\frac{2\sqrt{22}+4}{3}$$

The equation is now solved.

$$x=\frac{4-2\sqrt{22}}{3} x=\frac{2\sqrt{22}+4}{3}$$

Steps for Completing the Square

Subtract x^{2} from both sides.

$$-2x^{2}+6x-x^{2}=-2x-24$$

Combine -2x^{2} and -x^{2} to get -3x^{2}.

$$-3x^{2}+6x=-2x-24$$

Add 2x to both sides.

$$-3x^{2}+6x+2x=-24$$

Combine 6x and 2x to get 8x.

$$-3x^{2}+8x=-24$$

Divide both sides by -3.

$$\frac{-3x^{2}+8x}{-3}=-\frac{24}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\frac{8}{-3}x=-\frac{24}{-3}$$

Divide 8 by -3.

$$x^{2}-\frac{8}{3}x=-\frac{24}{-3}$$

Divide -24 by -3.

$$x^{2}-\frac{8}{3}x=8$$

Divide -\frac{8}{3}, the coefficient of the x term, by 2 to get -\frac{4}{3}. Then add the square of -\frac{4}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{8}{3}x+\left(-\frac{4}{3}\right)^{2}=8+\left(-\frac{4}{3}\right)^{2}$$

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

$$x^{2}-\frac{8}{3}x+\frac{16}{9}=8+\frac{16}{9}$$

Add 8 to \frac{16}{9}.

$$x^{2}-\frac{8}{3}x+\frac{16}{9}=\frac{88}{9}$$

Factor x^{2}-\frac{8}{3}x+\frac{16}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{4}{3}\right)^{2}=\frac{88}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{4}{3}\right)^{2}}=\sqrt{\frac{88}{9}}$$

Simplify.

$$x-\frac{4}{3}=\frac{2\sqrt{22}}{3} x-\frac{4}{3}=-\frac{2\sqrt{22}}{3}$$

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

$$x=\frac{2\sqrt{22}+4}{3} x=\frac{4-2\sqrt{22}}{3}$$