Solve 72x=8x^2-1552

$$72 x = 8 x ^ { 2 } - 1552$$

Solve for x

$$x = \frac{\sqrt{857} + 9}{2} \approx 19.137281168$$

$$x=\frac{9-\sqrt{857}}{2}\approx -10.137281168$$

Steps Using the Quadratic Formula

Subtract 8x^{2} from both sides.

$$72x-8x^{2}=-1552$$

Add 1552 to both sides.

$$72x-8x^{2}+1552=0$$

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.

$$-8x^{2}+72x+1552=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -8 for a, 72 for b, and 1552 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-72±\sqrt{72^{2}-4\left(-8\right)\times 1552}}{2\left(-8\right)}$$

Square 72.

$$x=\frac{-72±\sqrt{5184-4\left(-8\right)\times 1552}}{2\left(-8\right)}$$

Multiply -4 times -8.

$$x=\frac{-72±\sqrt{5184+32\times 1552}}{2\left(-8\right)}$$

Multiply 32 times 1552.

$$x=\frac{-72±\sqrt{5184+49664}}{2\left(-8\right)}$$

Add 5184 to 49664.

$$x=\frac{-72±\sqrt{54848}}{2\left(-8\right)}$$

Take the square root of 54848.

$$x=\frac{-72±8\sqrt{857}}{2\left(-8\right)}$$

Multiply 2 times -8.

$$x=\frac{-72±8\sqrt{857}}{-16}$$

Now solve the equation x=\frac{-72±8\sqrt{857}}{-16} when ± is plus. Add -72 to 8\sqrt{857}.

$$x=\frac{8\sqrt{857}-72}{-16}$$

Divide -72+8\sqrt{857} by -16.

$$x=\frac{9-\sqrt{857}}{2}$$

Now solve the equation x=\frac{-72±8\sqrt{857}}{-16} when ± is minus. Subtract 8\sqrt{857} from -72.

$$x=\frac{-8\sqrt{857}-72}{-16}$$

Divide -72-8\sqrt{857} by -16.

$$x=\frac{\sqrt{857}+9}{2}$$

The equation is now solved.

$$x=\frac{9-\sqrt{857}}{2} x=\frac{\sqrt{857}+9}{2}$$

Steps for Completing the Square

Subtract 8x^{2} from both sides.

$$72x-8x^{2}=-1552$$

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.

$$-8x^{2}+72x=-1552$$

Divide both sides by -8.

$$\frac{-8x^{2}+72x}{-8}=-\frac{1552}{-8}$$

Dividing by -8 undoes the multiplication by -8.

$$x^{2}+\frac{72}{-8}x=-\frac{1552}{-8}$$

Divide 72 by -8.

$$x^{2}-9x=-\frac{1552}{-8}$$

Divide -1552 by -8.

$$x^{2}-9x=194$$

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

$$x^{2}-9x+\left(-\frac{9}{2}\right)^{2}=194+\left(-\frac{9}{2}\right)^{2}$$

Square -\frac{9}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-9x+\frac{81}{4}=194+\frac{81}{4}$$

Add 194 to \frac{81}{4}.

$$x^{2}-9x+\frac{81}{4}=\frac{857}{4}$$

Factor x^{2}-9x+\frac{81}{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}.

$$\left(x-\frac{9}{2}\right)^{2}=\frac{857}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{2}\right)^{2}}=\sqrt{\frac{857}{4}}$$

Simplify.

$$x-\frac{9}{2}=\frac{\sqrt{857}}{2} x-\frac{9}{2}=-\frac{\sqrt{857}}{2}$$

Add \frac{9}{2} to both sides of the equation.

$$x=\frac{\sqrt{857}+9}{2} x=\frac{9-\sqrt{857}}{2}$$