Solve 85x+9x^2div79=2

$$85x+9 { x }^{ 2 } \div 79=2$$

Solve for x

$$x=\frac{\sqrt{45096913}-6715}{18}\approx 0.02352867$$

$$x=\frac{-\sqrt{45096913}-6715}{18}\approx -746.134639781$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 79.

$$6715x+9x^{2}=158$$

Subtract 158 from both sides.

$$6715x+9x^{2}-158=0$$

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.

$$9x^{2}+6715x-158=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, 6715 for b, and -158 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6715±\sqrt{6715^{2}-4\times 9\left(-158\right)}}{2\times 9}$$

Square 6715.

$$x=\frac{-6715±\sqrt{45091225-4\times 9\left(-158\right)}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-6715±\sqrt{45091225-36\left(-158\right)}}{2\times 9}$$

Multiply -36 times -158.

$$x=\frac{-6715±\sqrt{45091225+5688}}{2\times 9}$$

Add 45091225 to 5688.

$$x=\frac{-6715±\sqrt{45096913}}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{-6715±\sqrt{45096913}}{18}$$

Now solve the equation x=\frac{-6715±\sqrt{45096913}}{18} when ± is plus. Add -6715 to \sqrt{45096913}.

$$x=\frac{\sqrt{45096913}-6715}{18}$$

Now solve the equation x=\frac{-6715±\sqrt{45096913}}{18} when ± is minus. Subtract \sqrt{45096913} from -6715.

$$x=\frac{-\sqrt{45096913}-6715}{18}$$

The equation is now solved.

$$x=\frac{\sqrt{45096913}-6715}{18} x=\frac{-\sqrt{45096913}-6715}{18}$$

Steps for Completing the Square

Multiply both sides of the equation by 79.

$$6715x+9x^{2}=158$$

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.

$$9x^{2}+6715x=158$$

Divide both sides by 9.

$$\frac{9x^{2}+6715x}{9}=\frac{158}{9}$$

Dividing by 9 undoes the multiplication by 9.

$$x^{2}+\frac{6715}{9}x=\frac{158}{9}$$

Divide \frac{6715}{9}, the coefficient of the x term, by 2 to get \frac{6715}{18}. Then add the square of \frac{6715}{18} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{6715}{9}x+\left(\frac{6715}{18}\right)^{2}=\frac{158}{9}+\left(\frac{6715}{18}\right)^{2}$$

Square \frac{6715}{18} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{6715}{9}x+\frac{45091225}{324}=\frac{158}{9}+\frac{45091225}{324}$$

Add \frac{158}{9} to \frac{45091225}{324} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{6715}{9}x+\frac{45091225}{324}=\frac{45096913}{324}$$

Factor x^{2}+\frac{6715}{9}x+\frac{45091225}{324}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{6715}{18}\right)^{2}=\frac{45096913}{324}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{6715}{18}\right)^{2}}=\sqrt{\frac{45096913}{324}}$$

Simplify.

$$x+\frac{6715}{18}=\frac{\sqrt{45096913}}{18} x+\frac{6715}{18}=-\frac{\sqrt{45096913}}{18}$$

Subtract \frac{6715}{18} from both sides of the equation.

$$x=\frac{\sqrt{45096913}-6715}{18} x=\frac{-\sqrt{45096913}-6715}{18}$$