Solve -x^2+4x-72=-2x^2+8

$$- x ^ { 2 } + 4 x - 72 = - 2 x ^ { 2 } + 8$$

Solve for x

$$x=2\sqrt{21}-2\approx 7.16515139$$

$$x=-2\sqrt{21}-2\approx -11.16515139$$

Steps Using the Quadratic Formula

Add 2x^{2} to both sides.

$$-x^{2}+4x-72+2x^{2}=8$$

Subtract 8 from both sides.

$$-x^{2}+4x-72+2x^{2}-8=0$$

Subtract 8 from -72 to get -80.

$$-x^{2}+4x-80+2x^{2}=0$$

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

$$x^{2}+4x-80=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 4 for b, and -80 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\left(-80\right)}}{2}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\left(-80\right)}}{2}$$

Multiply -4 times -80.

$$x=\frac{-4±\sqrt{16+320}}{2}$$

Add 16 to 320.

$$x=\frac{-4±\sqrt{336}}{2}$$

Take the square root of 336.

$$x=\frac{-4±4\sqrt{21}}{2}$$

Now solve the equation x=\frac{-4±4\sqrt{21}}{2} when ± is plus. Add -4 to 4\sqrt{21}.

$$x=\frac{4\sqrt{21}-4}{2}$$

Divide -4+4\sqrt{21} by 2.

$$x=2\sqrt{21}-2$$

Now solve the equation x=\frac{-4±4\sqrt{21}}{2} when ± is minus. Subtract 4\sqrt{21} from -4.

$$x=\frac{-4\sqrt{21}-4}{2}$$

Divide -4-4\sqrt{21} by 2.

$$x=-2\sqrt{21}-2$$

The equation is now solved.

$$x=2\sqrt{21}-2 x=-2\sqrt{21}-2$$

Steps for Completing the Square

Add 2x^{2} to both sides.

$$-x^{2}+4x-72+2x^{2}=8$$

Add 72 to both sides.

$$-x^{2}+4x+2x^{2}=8+72$$

Add 8 and 72 to get 80.

$$-x^{2}+4x+2x^{2}=80$$

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

$$x^{2}+4x=80$$

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

$$x^{2}+4x+2^{2}=80+2^{2}$$

Square 2.

$$x^{2}+4x+4=80+4$$

Add 80 to 4.

$$x^{2}+4x+4=84$$

Factor x^{2}+4x+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+2\right)^{2}=84$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{84}$$

Simplify.

$$x+2=2\sqrt{21} x+2=-2\sqrt{21}$$

Subtract 2 from both sides of the equation.

$$x=2\sqrt{21}-2 x=-2\sqrt{21}-2$$