Solve 320=4x^2-68x^2+288x

$$320 = 4 { x }^{ 2 } -68 { x }^{ 2 } +288x$$

Solve for x

$$x=2$$

$$x = \frac{5}{2} = 2\frac{1}{2} = 2.5$$

Steps Using the Quadratic Formula

Combine 4x^{2} and -68x^{2} to get -64x^{2}.

$$320=-64x^{2}+288x$$

Swap sides so that all variable terms are on the left hand side.

$$-64x^{2}+288x=320$$

Subtract 320 from both sides.

$$-64x^{2}+288x-320=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -64 for a, 288 for b, and -320 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-288±\sqrt{288^{2}-4\left(-64\right)\left(-320\right)}}{2\left(-64\right)}$$

Square 288.

$$x=\frac{-288±\sqrt{82944-4\left(-64\right)\left(-320\right)}}{2\left(-64\right)}$$

Multiply -4 times -64.

$$x=\frac{-288±\sqrt{82944+256\left(-320\right)}}{2\left(-64\right)}$$

Multiply 256 times -320.

$$x=\frac{-288±\sqrt{82944-81920}}{2\left(-64\right)}$$

Add 82944 to -81920.

$$x=\frac{-288±\sqrt{1024}}{2\left(-64\right)}$$

Take the square root of 1024.

$$x=\frac{-288±32}{2\left(-64\right)}$$

Multiply 2 times -64.

$$x=\frac{-288±32}{-128}$$

Now solve the equation x=\frac{-288±32}{-128} when ± is plus. Add -288 to 32.

$$x=-\frac{256}{-128}$$

Divide -256 by -128.

$$x=2$$

Now solve the equation x=\frac{-288±32}{-128} when ± is minus. Subtract 32 from -288.

$$x=-\frac{320}{-128}$$

Reduce the fraction \frac{-320}{-128} to lowest terms by extracting and canceling out 64.

$$x=\frac{5}{2}$$

The equation is now solved.

$$x=2 x=\frac{5}{2}$$

Steps for Completing the Square

Combine 4x^{2} and -68x^{2} to get -64x^{2}.

$$320=-64x^{2}+288x$$

Swap sides so that all variable terms are on the left hand side.

$$-64x^{2}+288x=320$$

Divide both sides by -64.

$$\frac{-64x^{2}+288x}{-64}=\frac{320}{-64}$$

Dividing by -64 undoes the multiplication by -64.

$$x^{2}+\frac{288}{-64}x=\frac{320}{-64}$$

Reduce the fraction \frac{288}{-64} to lowest terms by extracting and canceling out 32.

$$x^{2}-\frac{9}{2}x=\frac{320}{-64}$$

Divide 320 by -64.

$$x^{2}-\frac{9}{2}x=-5$$

Divide -\frac{9}{2}, the coefficient of the x term, by 2 to get -\frac{9}{4}. Then add the square of -\frac{9}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{9}{2}x+\left(-\frac{9}{4}\right)^{2}=-5+\left(-\frac{9}{4}\right)^{2}$$

Square -\frac{9}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{9}{2}x+\frac{81}{16}=-5+\frac{81}{16}$$

Add -5 to \frac{81}{16}.

$$x^{2}-\frac{9}{2}x+\frac{81}{16}=\frac{1}{16}$$

Factor x^{2}-\frac{9}{2}x+\frac{81}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{9}{4}\right)^{2}=\frac{1}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{4}\right)^{2}}=\sqrt{\frac{1}{16}}$$

Simplify.

$$x-\frac{9}{4}=\frac{1}{4} x-\frac{9}{4}=-\frac{1}{4}$$

Add \frac{9}{4} to both sides of the equation.

$$x=\frac{5}{2} x=2$$