Solve 1707x^2+960x-5225=0

$$1707 { x }^{ 2 } +960x-5225=0$$

Solve for x

$$x=\frac{5\sqrt{365979}}{1707}-\frac{160}{569}\approx 1.490808384$$

$$x=-\frac{5\sqrt{365979}}{1707}-\frac{160}{569}\approx -2.053198542$$

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.

$$1707x^{2}+960x-5225=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1707 for a, 960 for b, and -5225 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-960±\sqrt{960^{2}-4\times 1707\left(-5225\right)}}{2\times 1707}$$

Square 960.

$$x=\frac{-960±\sqrt{921600-4\times 1707\left(-5225\right)}}{2\times 1707}$$

Multiply -4 times 1707.

$$x=\frac{-960±\sqrt{921600-6828\left(-5225\right)}}{2\times 1707}$$

Multiply -6828 times -5225.

$$x=\frac{-960±\sqrt{921600+35676300}}{2\times 1707}$$

Add 921600 to 35676300.

$$x=\frac{-960±\sqrt{36597900}}{2\times 1707}$$

Take the square root of 36597900.

$$x=\frac{-960±10\sqrt{365979}}{2\times 1707}$$

Multiply 2 times 1707.

$$x=\frac{-960±10\sqrt{365979}}{3414}$$

Now solve the equation x=\frac{-960±10\sqrt{365979}}{3414} when ± is plus. Add -960 to 10\sqrt{365979}.

$$x=\frac{10\sqrt{365979}-960}{3414}$$

Divide -960+10\sqrt{365979} by 3414.

$$x=\frac{5\sqrt{365979}}{1707}-\frac{160}{569}$$

Now solve the equation x=\frac{-960±10\sqrt{365979}}{3414} when ± is minus. Subtract 10\sqrt{365979} from -960.

$$x=\frac{-10\sqrt{365979}-960}{3414}$$

Divide -960-10\sqrt{365979} by 3414.

$$x=-\frac{5\sqrt{365979}}{1707}-\frac{160}{569}$$

The equation is now solved.

$$x=\frac{5\sqrt{365979}}{1707}-\frac{160}{569} x=-\frac{5\sqrt{365979}}{1707}-\frac{160}{569}$$

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.

$$1707x^{2}+960x-5225=0$$

Add 5225 to both sides of the equation.

$$1707x^{2}+960x-5225-\left(-5225\right)=-\left(-5225\right)$$

Subtracting -5225 from itself leaves 0.

$$1707x^{2}+960x=-\left(-5225\right)$$

Subtract -5225 from 0.

$$1707x^{2}+960x=5225$$

Divide both sides by 1707.

$$\frac{1707x^{2}+960x}{1707}=\frac{5225}{1707}$$

Dividing by 1707 undoes the multiplication by 1707.

$$x^{2}+\frac{960}{1707}x=\frac{5225}{1707}$$

Reduce the fraction \frac{960}{1707} to lowest terms by extracting and canceling out 3.

$$x^{2}+\frac{320}{569}x=\frac{5225}{1707}$$

Divide \frac{320}{569}, the coefficient of the x term, by 2 to get \frac{160}{569}. Then add the square of \frac{160}{569} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{320}{569}x+\left(\frac{160}{569}\right)^{2}=\frac{5225}{1707}+\left(\frac{160}{569}\right)^{2}$$

Square \frac{160}{569} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{320}{569}x+\frac{25600}{323761}=\frac{5225}{1707}+\frac{25600}{323761}$$

Add \frac{5225}{1707} to \frac{25600}{323761} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{320}{569}x+\frac{25600}{323761}=\frac{3049825}{971283}$$

Factor x^{2}+\frac{320}{569}x+\frac{25600}{323761}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{160}{569}\right)^{2}=\frac{3049825}{971283}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{160}{569}\right)^{2}}=\sqrt{\frac{3049825}{971283}}$$

Simplify.

$$x+\frac{160}{569}=\frac{5\sqrt{365979}}{1707} x+\frac{160}{569}=-\frac{5\sqrt{365979}}{1707}$$

Subtract \frac{160}{569} from both sides of the equation.

$$x=\frac{5\sqrt{365979}}{1707}-\frac{160}{569} x=-\frac{5\sqrt{365979}}{1707}-\frac{160}{569}$$