Solve 240x=240-16x^2

$$240x = 240-16 { x }^{ 2 }$$

Solve for x

$$x=\frac{\sqrt{285}-15}{2}\approx 0.940971508$$

$$x=\frac{-\sqrt{285}-15}{2}\approx -15.940971508$$

Steps Using the Quadratic Formula

Subtract 240 from both sides.

$$240x-240=-16x^{2}$$

Add 16x^{2} to both sides.

$$240x-240+16x^{2}=0$$

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.

$$16x^{2}+240x-240=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 16 for a, 240 for b, and -240 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-240±\sqrt{240^{2}-4\times 16\left(-240\right)}}{2\times 16}$$

Square 240.

$$x=\frac{-240±\sqrt{57600-4\times 16\left(-240\right)}}{2\times 16}$$

Multiply -4 times 16.

$$x=\frac{-240±\sqrt{57600-64\left(-240\right)}}{2\times 16}$$

Multiply -64 times -240.

$$x=\frac{-240±\sqrt{57600+15360}}{2\times 16}$$

Add 57600 to 15360.

$$x=\frac{-240±\sqrt{72960}}{2\times 16}$$

Take the square root of 72960.

$$x=\frac{-240±16\sqrt{285}}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{-240±16\sqrt{285}}{32}$$

Now solve the equation x=\frac{-240±16\sqrt{285}}{32} when ± is plus. Add -240 to 16\sqrt{285}.

$$x=\frac{16\sqrt{285}-240}{32}$$

Divide -240+16\sqrt{285} by 32.

$$x=\frac{\sqrt{285}-15}{2}$$

Now solve the equation x=\frac{-240±16\sqrt{285}}{32} when ± is minus. Subtract 16\sqrt{285} from -240.

$$x=\frac{-16\sqrt{285}-240}{32}$$

Divide -240-16\sqrt{285} by 32.

$$x=\frac{-\sqrt{285}-15}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{285}-15}{2} x=\frac{-\sqrt{285}-15}{2}$$

Steps for Completing the Square

Add 16x^{2} to both sides.

$$240x+16x^{2}=240$$

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.

$$16x^{2}+240x=240$$

Divide both sides by 16.

$$\frac{16x^{2}+240x}{16}=\frac{240}{16}$$

Dividing by 16 undoes the multiplication by 16.

$$x^{2}+\frac{240}{16}x=\frac{240}{16}$$

Divide 240 by 16.

$$x^{2}+15x=\frac{240}{16}$$

Divide 240 by 16.

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

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.

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

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

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

Add 15 to \frac{225}{4}.

$$x^{2}+15x+\frac{225}{4}=\frac{285}{4}$$

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}.

$$\left(x+\frac{15}{2}\right)^{2}=\frac{285}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{15}{2}\right)^{2}}=\sqrt{\frac{285}{4}}$$

Simplify.

$$x+\frac{15}{2}=\frac{\sqrt{285}}{2} x+\frac{15}{2}=-\frac{\sqrt{285}}{2}$$

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

$$x=\frac{\sqrt{285}-15}{2} x=\frac{-\sqrt{285}-15}{2}$$