Solve 80x^2+96x-1400=0

$$80 { x }^{ 2 } +96x-1400=0$$

Solve for x

$$x=\frac{\sqrt{1786}}{10}-\frac{3}{5}\approx 3.626109322$$

$$x=-\frac{\sqrt{1786}}{10}-\frac{3}{5}\approx -4.826109322$$

Steps Using the Quadratic Formula

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.

$$80x^{2}+96x-1400=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 80 for a, 96 for b, and -1400 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-96±\sqrt{96^{2}-4\times 80\left(-1400\right)}}{2\times 80}$$

Square 96.

$$x=\frac{-96±\sqrt{9216-4\times 80\left(-1400\right)}}{2\times 80}$$

Multiply -4 times 80.

$$x=\frac{-96±\sqrt{9216-320\left(-1400\right)}}{2\times 80}$$

Multiply -320 times -1400.

$$x=\frac{-96±\sqrt{9216+448000}}{2\times 80}$$

Add 9216 to 448000.

$$x=\frac{-96±\sqrt{457216}}{2\times 80}$$

Take the square root of 457216.

$$x=\frac{-96±16\sqrt{1786}}{2\times 80}$$

Multiply 2 times 80.

$$x=\frac{-96±16\sqrt{1786}}{160}$$

Now solve the equation x=\frac{-96±16\sqrt{1786}}{160} when ± is plus. Add -96 to 16\sqrt{1786}.

$$x=\frac{16\sqrt{1786}-96}{160}$$

Divide -96+16\sqrt{1786} by 160.

$$x=\frac{\sqrt{1786}}{10}-\frac{3}{5}$$

Now solve the equation x=\frac{-96±16\sqrt{1786}}{160} when ± is minus. Subtract 16\sqrt{1786} from -96.

$$x=\frac{-16\sqrt{1786}-96}{160}$$

Divide -96-16\sqrt{1786} by 160.

$$x=-\frac{\sqrt{1786}}{10}-\frac{3}{5}$$

The equation is now solved.

$$x=\frac{\sqrt{1786}}{10}-\frac{3}{5} x=-\frac{\sqrt{1786}}{10}-\frac{3}{5}$$

Steps for Completing the Square

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.

$$80x^{2}+96x-1400=0$$

Add 1400 to both sides of the equation.

$$80x^{2}+96x-1400-\left(-1400\right)=-\left(-1400\right)$$

Subtracting -1400 from itself leaves 0.

$$80x^{2}+96x=-\left(-1400\right)$$

Subtract -1400 from 0.

$$80x^{2}+96x=1400$$

Divide both sides by 80.

$$\frac{80x^{2}+96x}{80}=\frac{1400}{80}$$

Dividing by 80 undoes the multiplication by 80.

$$x^{2}+\frac{96}{80}x=\frac{1400}{80}$$

Reduce the fraction \frac{96}{80} to lowest terms by extracting and canceling out 16.

$$x^{2}+\frac{6}{5}x=\frac{1400}{80}$$

Reduce the fraction \frac{1400}{80} to lowest terms by extracting and canceling out 40.

$$x^{2}+\frac{6}{5}x=\frac{35}{2}$$

Divide \frac{6}{5}, the coefficient of the x term, by 2 to get \frac{3}{5}. Then add the square of \frac{3}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{6}{5}x+\left(\frac{3}{5}\right)^{2}=\frac{35}{2}+\left(\frac{3}{5}\right)^{2}$$

Square \frac{3}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{6}{5}x+\frac{9}{25}=\frac{35}{2}+\frac{9}{25}$$

Add \frac{35}{2} to \frac{9}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{6}{5}x+\frac{9}{25}=\frac{893}{50}$$

Factor x^{2}+\frac{6}{5}x+\frac{9}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{3}{5}\right)^{2}=\frac{893}{50}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{5}\right)^{2}}=\sqrt{\frac{893}{50}}$$

Simplify.

$$x+\frac{3}{5}=\frac{\sqrt{1786}}{10} x+\frac{3}{5}=-\frac{\sqrt{1786}}{10}$$

Subtract \frac{3}{5} from both sides of the equation.

$$x=\frac{\sqrt{1786}}{10}-\frac{3}{5} x=-\frac{\sqrt{1786}}{10}-\frac{3}{5}$$