Solve 5x^2+14x(15-5x)+9=108

$$5 { x }^{ 2 } +14x(15-5x)+9=108$$

Solve for x

$$x=\frac{3\sqrt{510}}{65}+\frac{21}{13}\approx 2.657685211$$

$$x=-\frac{3\sqrt{510}}{65}+\frac{21}{13}\approx 0.573084019$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 14x by 15-5x.

$$5x^{2}+210x-70x^{2}+9=108$$

Combine 5x^{2} and -70x^{2} to get -65x^{2}.

$$-65x^{2}+210x+9=108$$

Subtract 108 from both sides.

$$-65x^{2}+210x+9-108=0$$

Subtract 108 from 9 to get -99.

$$-65x^{2}+210x-99=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -65 for a, 210 for b, and -99 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-210±\sqrt{210^{2}-4\left(-65\right)\left(-99\right)}}{2\left(-65\right)}$$

Square 210.

$$x=\frac{-210±\sqrt{44100-4\left(-65\right)\left(-99\right)}}{2\left(-65\right)}$$

Multiply -4 times -65.

$$x=\frac{-210±\sqrt{44100+260\left(-99\right)}}{2\left(-65\right)}$$

Multiply 260 times -99.

$$x=\frac{-210±\sqrt{44100-25740}}{2\left(-65\right)}$$

Add 44100 to -25740.

$$x=\frac{-210±\sqrt{18360}}{2\left(-65\right)}$$

Take the square root of 18360.

$$x=\frac{-210±6\sqrt{510}}{2\left(-65\right)}$$

Multiply 2 times -65.

$$x=\frac{-210±6\sqrt{510}}{-130}$$

Now solve the equation x=\frac{-210±6\sqrt{510}}{-130} when ± is plus. Add -210 to 6\sqrt{510}.

$$x=\frac{6\sqrt{510}-210}{-130}$$

Divide -210+6\sqrt{510} by -130.

$$x=-\frac{3\sqrt{510}}{65}+\frac{21}{13}$$

Now solve the equation x=\frac{-210±6\sqrt{510}}{-130} when ± is minus. Subtract 6\sqrt{510} from -210.

$$x=\frac{-6\sqrt{510}-210}{-130}$$

Divide -210-6\sqrt{510} by -130.

$$x=\frac{3\sqrt{510}}{65}+\frac{21}{13}$$

The equation is now solved.

$$x=-\frac{3\sqrt{510}}{65}+\frac{21}{13} x=\frac{3\sqrt{510}}{65}+\frac{21}{13}$$

Steps for Completing the Square

Use the distributive property to multiply 14x by 15-5x.

$$5x^{2}+210x-70x^{2}+9=108$$

Combine 5x^{2} and -70x^{2} to get -65x^{2}.

$$-65x^{2}+210x+9=108$$

Subtract 9 from both sides.

$$-65x^{2}+210x=108-9$$

Subtract 9 from 108 to get 99.

$$-65x^{2}+210x=99$$

Divide both sides by -65.

$$\frac{-65x^{2}+210x}{-65}=\frac{99}{-65}$$

Dividing by -65 undoes the multiplication by -65.

$$x^{2}+\frac{210}{-65}x=\frac{99}{-65}$$

Reduce the fraction \frac{210}{-65} to lowest terms by extracting and canceling out 5.

$$x^{2}-\frac{42}{13}x=\frac{99}{-65}$$

Divide 99 by -65.

$$x^{2}-\frac{42}{13}x=-\frac{99}{65}$$

Divide -\frac{42}{13}, the coefficient of the x term, by 2 to get -\frac{21}{13}. Then add the square of -\frac{21}{13} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{42}{13}x+\left(-\frac{21}{13}\right)^{2}=-\frac{99}{65}+\left(-\frac{21}{13}\right)^{2}$$

Square -\frac{21}{13} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{42}{13}x+\frac{441}{169}=-\frac{99}{65}+\frac{441}{169}$$

Add -\frac{99}{65} to \frac{441}{169} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{42}{13}x+\frac{441}{169}=\frac{918}{845}$$

Factor x^{2}-\frac{42}{13}x+\frac{441}{169}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{21}{13}\right)^{2}=\frac{918}{845}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{21}{13}\right)^{2}}=\sqrt{\frac{918}{845}}$$

Simplify.

$$x-\frac{21}{13}=\frac{3\sqrt{510}}{65} x-\frac{21}{13}=-\frac{3\sqrt{510}}{65}$$

Add \frac{21}{13} to both sides of the equation.

$$x=\frac{3\sqrt{510}}{65}+\frac{21}{13} x=-\frac{3\sqrt{510}}{65}+\frac{21}{13}$$