Solve 100x-40x+4x^2-72=0

$$100 x - 40 x + 4 x ^ { 2 } - 72 = 0$$

Solve for x

$$x = \frac{3 \sqrt{33} - 15}{2} \approx 1.11684397$$

$$x=\frac{-3\sqrt{33}-15}{2}\approx -16.11684397$$

Steps Using the Quadratic Formula

Combine 100x and -40x to get 60x.

$$60x+4x^{2}-72=0$$

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.

$$4x^{2}+60x-72=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 60 for b, and -72 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-60±\sqrt{60^{2}-4\times 4\left(-72\right)}}{2\times 4}$$

Square 60.

$$x=\frac{-60±\sqrt{3600-4\times 4\left(-72\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-60±\sqrt{3600-16\left(-72\right)}}{2\times 4}$$

Multiply -16 times -72.

$$x=\frac{-60±\sqrt{3600+1152}}{2\times 4}$$

Add 3600 to 1152.

$$x=\frac{-60±\sqrt{4752}}{2\times 4}$$

Take the square root of 4752.

$$x=\frac{-60±12\sqrt{33}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-60±12\sqrt{33}}{8}$$

Now solve the equation x=\frac{-60±12\sqrt{33}}{8} when ± is plus. Add -60 to 12\sqrt{33}.

$$x=\frac{12\sqrt{33}-60}{8}$$

Divide -60+12\sqrt{33} by 8.

$$x=\frac{3\sqrt{33}-15}{2}$$

Now solve the equation x=\frac{-60±12\sqrt{33}}{8} when ± is minus. Subtract 12\sqrt{33} from -60.

$$x=\frac{-12\sqrt{33}-60}{8}$$

Divide -60-12\sqrt{33} by 8.

$$x=\frac{-3\sqrt{33}-15}{2}$$

The equation is now solved.

$$x=\frac{3\sqrt{33}-15}{2} x=\frac{-3\sqrt{33}-15}{2}$$

Steps for Completing the Square

Combine 100x and -40x to get 60x.

$$60x+4x^{2}-72=0$$

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

$$60x+4x^{2}=72$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$4x^{2}+60x=72$$

Divide both sides by 4.

$$\frac{4x^{2}+60x}{4}=\frac{72}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\frac{60}{4}x=\frac{72}{4}$$

Divide 60 by 4.

$$x^{2}+15x=\frac{72}{4}$$

Divide 72 by 4.

$$x^{2}+15x=18$$

Divide 15, the coefficient of the x term, by 2 to get \frac{15}{2}. Then add the square of \frac{15}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+15x+\left(\frac{15}{2}\right)^{2}=18+\left(\frac{15}{2}\right)^{2}$$

Square \frac{15}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+15x+\frac{225}{4}=18+\frac{225}{4}$$

Add 18 to \frac{225}{4}.

$$x^{2}+15x+\frac{225}{4}=\frac{297}{4}$$

Factor x^{2}+15x+\frac{225}{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}.

$$\left(x+\frac{15}{2}\right)^{2}=\frac{297}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{15}{2}\right)^{2}}=\sqrt{\frac{297}{4}}$$

Simplify.

$$x+\frac{15}{2}=\frac{3\sqrt{33}}{2} x+\frac{15}{2}=-\frac{3\sqrt{33}}{2}$$

Subtract \frac{15}{2} from both sides of the equation.

$$x=\frac{3\sqrt{33}-15}{2} x=\frac{-3\sqrt{33}-15}{2}$$