Solve 8x+6x(11-x)=100

$$8 x + 6 x ( 11 - x ) = 100$$

Solve for x

$$x = \frac{\sqrt{769} + 37}{6} \approx 10.788474875$$

$$x = \frac{37 - \sqrt{769}}{6} \approx 1.544858459$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 6x by 11-x.

$$8x+66x-6x^{2}=100$$

Combine 8x and 66x to get 74x.

$$74x-6x^{2}=100$$

Subtract 100 from both sides.

$$74x-6x^{2}-100=0$$

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.

$$-6x^{2}+74x-100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -6 for a, 74 for b, and -100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-74±\sqrt{74^{2}-4\left(-6\right)\left(-100\right)}}{2\left(-6\right)}$$

Square 74.

$$x=\frac{-74±\sqrt{5476-4\left(-6\right)\left(-100\right)}}{2\left(-6\right)}$$

Multiply -4 times -6.

$$x=\frac{-74±\sqrt{5476+24\left(-100\right)}}{2\left(-6\right)}$$

Multiply 24 times -100.

$$x=\frac{-74±\sqrt{5476-2400}}{2\left(-6\right)}$$

Add 5476 to -2400.

$$x=\frac{-74±\sqrt{3076}}{2\left(-6\right)}$$

Take the square root of 3076.

$$x=\frac{-74±2\sqrt{769}}{2\left(-6\right)}$$

Multiply 2 times -6.

$$x=\frac{-74±2\sqrt{769}}{-12}$$

Now solve the equation x=\frac{-74±2\sqrt{769}}{-12} when ± is plus. Add -74 to 2\sqrt{769}.

$$x=\frac{2\sqrt{769}-74}{-12}$$

Divide -74+2\sqrt{769} by -12.

$$x=\frac{37-\sqrt{769}}{6}$$

Now solve the equation x=\frac{-74±2\sqrt{769}}{-12} when ± is minus. Subtract 2\sqrt{769} from -74.

$$x=\frac{-2\sqrt{769}-74}{-12}$$

Divide -74-2\sqrt{769} by -12.

$$x=\frac{\sqrt{769}+37}{6}$$

The equation is now solved.

$$x=\frac{37-\sqrt{769}}{6} x=\frac{\sqrt{769}+37}{6}$$

Steps for Completing the Square

Use the distributive property to multiply 6x by 11-x.

$$8x+66x-6x^{2}=100$$

Combine 8x and 66x to get 74x.

$$74x-6x^{2}=100$$

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.

$$-6x^{2}+74x=100$$

Divide both sides by -6.

$$\frac{-6x^{2}+74x}{-6}=\frac{100}{-6}$$

Dividing by -6 undoes the multiplication by -6.

$$x^{2}+\frac{74}{-6}x=\frac{100}{-6}$$

Reduce the fraction \frac{74}{-6} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{37}{3}x=\frac{100}{-6}$$

Reduce the fraction \frac{100}{-6} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{37}{3}x=-\frac{50}{3}$$

Divide -\frac{37}{3}, the coefficient of the x term, by 2 to get -\frac{37}{6}. Then add the square of -\frac{37}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{37}{3}x+\left(-\frac{37}{6}\right)^{2}=-\frac{50}{3}+\left(-\frac{37}{6}\right)^{2}$$

Square -\frac{37}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{37}{3}x+\frac{1369}{36}=-\frac{50}{3}+\frac{1369}{36}$$

Add -\frac{50}{3} to \frac{1369}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{37}{3}x+\frac{1369}{36}=\frac{769}{36}$$

Factor x^{2}-\frac{37}{3}x+\frac{1369}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{37}{6}\right)^{2}=\frac{769}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{37}{6}\right)^{2}}=\sqrt{\frac{769}{36}}$$

Simplify.

$$x-\frac{37}{6}=\frac{\sqrt{769}}{6} x-\frac{37}{6}=-\frac{\sqrt{769}}{6}$$

Add \frac{37}{6} to both sides of the equation.

$$x=\frac{\sqrt{769}+37}{6} x=\frac{37-\sqrt{769}}{6}$$