Solve 4x^2-72=24x

$$4 x ^ { 2 } - 72 = 24 x$$

Solve for x

$$x=3\sqrt{3}+3\approx 8.196152423$$

$$x=3-3\sqrt{3}\approx -2.196152423$$

Steps Using the Quadratic Formula

Subtract 24x from both sides.

$$4x^{2}-72-24x=0$$

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.

$$4x^{2}-24x-72=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -24 for b, and -72 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 4\left(-72\right)}}{2\times 4}$$

Square -24.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\times 4\left(-72\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-24\right)±\sqrt{576-16\left(-72\right)}}{2\times 4}$$

Multiply -16 times -72.

$$x=\frac{-\left(-24\right)±\sqrt{576+1152}}{2\times 4}$$

Add 576 to 1152.

$$x=\frac{-\left(-24\right)±\sqrt{1728}}{2\times 4}$$

Take the square root of 1728.

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

The opposite of -24 is 24.

$$x=\frac{24±24\sqrt{3}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{24±24\sqrt{3}}{8}$$

Now solve the equation x=\frac{24±24\sqrt{3}}{8} when ± is plus. Add 24 to 24\sqrt{3}.

$$x=\frac{24\sqrt{3}+24}{8}$$

Divide 24+24\sqrt{3} by 8.

$$x=3\sqrt{3}+3$$

Now solve the equation x=\frac{24±24\sqrt{3}}{8} when ± is minus. Subtract 24\sqrt{3} from 24.

$$x=\frac{24-24\sqrt{3}}{8}$$

Divide 24-24\sqrt{3} by 8.

$$x=3-3\sqrt{3}$$

The equation is now solved.

$$x=3\sqrt{3}+3 x=3-3\sqrt{3}$$

Steps for Completing the Square

Subtract 24x from both sides.

$$4x^{2}-72-24x=0$$

Add 72 to both sides. Anything plus zero gives itself.

$$4x^{2}-24x=72$$

Divide both sides by 4.

$$\frac{4x^{2}-24x}{4}=\frac{72}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\left(-\frac{24}{4}\right)x=\frac{72}{4}$$

Divide -24 by 4.

$$x^{2}-6x=\frac{72}{4}$$

Divide 72 by 4.

$$x^{2}-6x=18$$

Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-6x+\left(-3\right)^{2}=18+\left(-3\right)^{2}$$

Square -3.

$$x^{2}-6x+9=18+9$$

Add 18 to 9.

$$x^{2}-6x+9=27$$

Factor x^{2}-6x+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}.

$$\left(x-3\right)^{2}=27$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{27}$$

Simplify.

$$x-3=3\sqrt{3} x-3=-3\sqrt{3}$$

Add 3 to both sides of the equation.

$$x=3\sqrt{3}+3 x=3-3\sqrt{3}$$