Solve 85x/93+x^2=108

$$\frac { 85 x } { 93 } + x ^ { 2 } = 108$$

Solve for x

$$x = \frac{\sqrt{3743593} - 85}{186} \approx 9.945358525$$

$$x=\frac{-\sqrt{3743593}-85}{186}\approx -10.859337019$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 93.

$$85x+93x^{2}=10044$$

Subtract 10044 from both sides.

$$85x+93x^{2}-10044=0$$

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.

$$93x^{2}+85x-10044=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 93 for a, 85 for b, and -10044 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-85±\sqrt{85^{2}-4\times 93\left(-10044\right)}}{2\times 93}$$

Square 85.

$$x=\frac{-85±\sqrt{7225-4\times 93\left(-10044\right)}}{2\times 93}$$

Multiply -4 times 93.

$$x=\frac{-85±\sqrt{7225-372\left(-10044\right)}}{2\times 93}$$

Multiply -372 times -10044.

$$x=\frac{-85±\sqrt{7225+3736368}}{2\times 93}$$

Add 7225 to 3736368.

$$x=\frac{-85±\sqrt{3743593}}{2\times 93}$$

Multiply 2 times 93.

$$x=\frac{-85±\sqrt{3743593}}{186}$$

Now solve the equation x=\frac{-85±\sqrt{3743593}}{186} when ± is plus. Add -85 to \sqrt{3743593}.

$$x=\frac{\sqrt{3743593}-85}{186}$$

Now solve the equation x=\frac{-85±\sqrt{3743593}}{186} when ± is minus. Subtract \sqrt{3743593} from -85.

$$x=\frac{-\sqrt{3743593}-85}{186}$$

The equation is now solved.

$$x=\frac{\sqrt{3743593}-85}{186} x=\frac{-\sqrt{3743593}-85}{186}$$

Steps for Completing the Square

Multiply both sides of the equation by 93.

$$85x+93x^{2}=10044$$

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.

$$93x^{2}+85x=10044$$

Divide both sides by 93.

$$\frac{93x^{2}+85x}{93}=\frac{10044}{93}$$

Dividing by 93 undoes the multiplication by 93.

$$x^{2}+\frac{85}{93}x=\frac{10044}{93}$$

Divide 10044 by 93.

$$x^{2}+\frac{85}{93}x=108$$

Divide \frac{85}{93}, the coefficient of the x term, by 2 to get \frac{85}{186}. Then add the square of \frac{85}{186} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{85}{93}x+\left(\frac{85}{186}\right)^{2}=108+\left(\frac{85}{186}\right)^{2}$$

Square \frac{85}{186} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{85}{93}x+\frac{7225}{34596}=108+\frac{7225}{34596}$$

Add 108 to \frac{7225}{34596}.

$$x^{2}+\frac{85}{93}x+\frac{7225}{34596}=\frac{3743593}{34596}$$

Factor x^{2}+\frac{85}{93}x+\frac{7225}{34596}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{85}{186}\right)^{2}=\frac{3743593}{34596}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{85}{186}\right)^{2}}=\sqrt{\frac{3743593}{34596}}$$

Simplify.

$$x+\frac{85}{186}=\frac{\sqrt{3743593}}{186} x+\frac{85}{186}=-\frac{\sqrt{3743593}}{186}$$

Subtract \frac{85}{186} from both sides of the equation.

$$x=\frac{\sqrt{3743593}-85}{186} x=\frac{-\sqrt{3743593}-85}{186}$$