Solve 301x^2-918x=256

$$301 x ^ { 2 } - 918 x = 256$$

Solve for x

$$x = \frac{\sqrt{287737} + 459}{301} \approx 3.307014029$$

$$x=\frac{459-\sqrt{287737}}{301}\approx -0.257180142$$

Steps Using the Quadratic Formula

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.

$$301x^{2}-918x=256$$

Subtract 256 from both sides of the equation.

$$301x^{2}-918x-256=256-256$$

Subtracting 256 from itself leaves 0.

$$301x^{2}-918x-256=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 301 for a, -918 for b, and -256 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-918\right)±\sqrt{\left(-918\right)^{2}-4\times 301\left(-256\right)}}{2\times 301}$$

Square -918.

$$x=\frac{-\left(-918\right)±\sqrt{842724-4\times 301\left(-256\right)}}{2\times 301}$$

Multiply -4 times 301.

$$x=\frac{-\left(-918\right)±\sqrt{842724-1204\left(-256\right)}}{2\times 301}$$

Multiply -1204 times -256.

$$x=\frac{-\left(-918\right)±\sqrt{842724+308224}}{2\times 301}$$

Add 842724 to 308224.

$$x=\frac{-\left(-918\right)±\sqrt{1150948}}{2\times 301}$$

Take the square root of 1150948.

$$x=\frac{-\left(-918\right)±2\sqrt{287737}}{2\times 301}$$

The opposite of -918 is 918.

$$x=\frac{918±2\sqrt{287737}}{2\times 301}$$

Multiply 2 times 301.

$$x=\frac{918±2\sqrt{287737}}{602}$$

Now solve the equation x=\frac{918±2\sqrt{287737}}{602} when ± is plus. Add 918 to 2\sqrt{287737}.

$$x=\frac{2\sqrt{287737}+918}{602}$$

Divide 918+2\sqrt{287737} by 602.

$$x=\frac{\sqrt{287737}+459}{301}$$

Now solve the equation x=\frac{918±2\sqrt{287737}}{602} when ± is minus. Subtract 2\sqrt{287737} from 918.

$$x=\frac{918-2\sqrt{287737}}{602}$$

Divide 918-2\sqrt{287737} by 602.

$$x=\frac{459-\sqrt{287737}}{301}$$

The equation is now solved.

$$x=\frac{\sqrt{287737}+459}{301} x=\frac{459-\sqrt{287737}}{301}$$

Steps for Completing the Square

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.

$$301x^{2}-918x=256$$

Divide both sides by 301.

$$\frac{301x^{2}-918x}{301}=\frac{256}{301}$$

Dividing by 301 undoes the multiplication by 301.

$$x^{2}-\frac{918}{301}x=\frac{256}{301}$$

Divide -\frac{918}{301}, the coefficient of the x term, by 2 to get -\frac{459}{301}. Then add the square of -\frac{459}{301} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{918}{301}x+\left(-\frac{459}{301}\right)^{2}=\frac{256}{301}+\left(-\frac{459}{301}\right)^{2}$$

Square -\frac{459}{301} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{918}{301}x+\frac{210681}{90601}=\frac{256}{301}+\frac{210681}{90601}$$

Add \frac{256}{301} to \frac{210681}{90601} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{918}{301}x+\frac{210681}{90601}=\frac{287737}{90601}$$

Factor x^{2}-\frac{918}{301}x+\frac{210681}{90601}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{459}{301}\right)^{2}=\frac{287737}{90601}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{459}{301}\right)^{2}}=\sqrt{\frac{287737}{90601}}$$

Simplify.

$$x-\frac{459}{301}=\frac{\sqrt{287737}}{301} x-\frac{459}{301}=-\frac{\sqrt{287737}}{301}$$

Add \frac{459}{301} to both sides of the equation.

$$x=\frac{\sqrt{287737}+459}{301} x=\frac{459-\sqrt{287737}}{301}$$