Solve 72-(x^2/2+x^2/2-24x+12)=20

$$72-( \frac{ { x }^{ 2 } }{ 2 } + \frac{ { x }^{ 2 } }{ 2 } -24x+12) = 20$$

Solve for x

$$x=2\sqrt{46}+12\approx 25.564659966$$

$$x=12-2\sqrt{46}\approx -1.564659966$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$144-2\left(\frac{x^{2}}{2}+\frac{x^{2}}{2}-24x+12\right)=40$$

Combine \frac{x^{2}}{2} and \frac{x^{2}}{2} to get x^{2}.

$$144-2\left(x^{2}-24x+12\right)=40$$

Use the distributive property to multiply -2 by x^{2}-24x+12.

$$144-2x^{2}+48x-24=40$$

Subtract 24 from 144 to get 120.

$$120-2x^{2}+48x=40$$

Subtract 40 from both sides.

$$120-2x^{2}+48x-40=0$$

Subtract 40 from 120 to get 80.

$$80-2x^{2}+48x=0$$

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.

$$-2x^{2}+48x+80=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 48 for b, and 80 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-48±\sqrt{48^{2}-4\left(-2\right)\times 80}}{2\left(-2\right)}$$

Square 48.

$$x=\frac{-48±\sqrt{2304-4\left(-2\right)\times 80}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-48±\sqrt{2304+8\times 80}}{2\left(-2\right)}$$

Multiply 8 times 80.

$$x=\frac{-48±\sqrt{2304+640}}{2\left(-2\right)}$$

Add 2304 to 640.

$$x=\frac{-48±\sqrt{2944}}{2\left(-2\right)}$$

Take the square root of 2944.

$$x=\frac{-48±8\sqrt{46}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-48±8\sqrt{46}}{-4}$$

Now solve the equation x=\frac{-48±8\sqrt{46}}{-4} when ± is plus. Add -48 to 8\sqrt{46}.

$$x=\frac{8\sqrt{46}-48}{-4}$$

Divide -48+8\sqrt{46} by -4.

$$x=12-2\sqrt{46}$$

Now solve the equation x=\frac{-48±8\sqrt{46}}{-4} when ± is minus. Subtract 8\sqrt{46} from -48.

$$x=\frac{-8\sqrt{46}-48}{-4}$$

Divide -48-8\sqrt{46} by -4.

$$x=2\sqrt{46}+12$$

The equation is now solved.

$$x=12-2\sqrt{46} x=2\sqrt{46}+12$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$144-2\left(\frac{x^{2}}{2}+\frac{x^{2}}{2}-24x+12\right)=40$$

Combine \frac{x^{2}}{2} and \frac{x^{2}}{2} to get x^{2}.

$$144-2\left(x^{2}-24x+12\right)=40$$

Use the distributive property to multiply -2 by x^{2}-24x+12.

$$144-2x^{2}+48x-24=40$$

Subtract 24 from 144 to get 120.

$$120-2x^{2}+48x=40$$

Subtract 120 from both sides.

$$-2x^{2}+48x=40-120$$

Subtract 120 from 40 to get -80.

$$-2x^{2}+48x=-80$$

Divide both sides by -2.

$$\frac{-2x^{2}+48x}{-2}=-\frac{80}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{48}{-2}x=-\frac{80}{-2}$$

Divide 48 by -2.

$$x^{2}-24x=-\frac{80}{-2}$$

Divide -80 by -2.

$$x^{2}-24x=40$$

Divide -24, the coefficient of the x term, by 2 to get -12. Then add the square of -12 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-24x+\left(-12\right)^{2}=40+\left(-12\right)^{2}$$

Square -12.

$$x^{2}-24x+144=40+144$$

Add 40 to 144.

$$x^{2}-24x+144=184$$

Factor x^{2}-24x+144. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-12\right)^{2}=184$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-12\right)^{2}}=\sqrt{184}$$

Simplify.

$$x-12=2\sqrt{46} x-12=-2\sqrt{46}$$

Add 12 to both sides of the equation.

$$x=2\sqrt{46}+12 x=12-2\sqrt{46}$$