Solve 578x^2+39x-117=6

$$578 x ^ { 2 } + 39 x - 117 = 6$$

Solve for x

$$x=\frac{\sqrt{285897}-39}{1156}\approx 0.428800493$$

$$x=\frac{-\sqrt{285897}-39}{1156}\approx -0.496274542$$

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.

$$578x^{2}+39x-117=6$$

Subtract 6 from both sides of the equation.

$$578x^{2}+39x-117-6=6-6$$

Subtracting 6 from itself leaves 0.

$$578x^{2}+39x-117-6=0$$

Subtract 6 from -117.

$$578x^{2}+39x-123=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 578 for a, 39 for b, and -123 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-39±\sqrt{39^{2}-4\times 578\left(-123\right)}}{2\times 578}$$

Square 39.

$$x=\frac{-39±\sqrt{1521-4\times 578\left(-123\right)}}{2\times 578}$$

Multiply -4 times 578.

$$x=\frac{-39±\sqrt{1521-2312\left(-123\right)}}{2\times 578}$$

Multiply -2312 times -123.

$$x=\frac{-39±\sqrt{1521+284376}}{2\times 578}$$

Add 1521 to 284376.

$$x=\frac{-39±\sqrt{285897}}{2\times 578}$$

Multiply 2 times 578.

$$x=\frac{-39±\sqrt{285897}}{1156}$$

Now solve the equation x=\frac{-39±\sqrt{285897}}{1156} when ± is plus. Add -39 to \sqrt{285897}.

$$x=\frac{\sqrt{285897}-39}{1156}$$

Now solve the equation x=\frac{-39±\sqrt{285897}}{1156} when ± is minus. Subtract \sqrt{285897} from -39.

$$x=\frac{-\sqrt{285897}-39}{1156}$$

The equation is now solved.

$$x=\frac{\sqrt{285897}-39}{1156} x=\frac{-\sqrt{285897}-39}{1156}$$

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.

$$578x^{2}+39x-117=6$$

Add 117 to both sides of the equation.

$$578x^{2}+39x-117-\left(-117\right)=6-\left(-117\right)$$

Subtracting -117 from itself leaves 0.

$$578x^{2}+39x=6-\left(-117\right)$$

Subtract -117 from 6.

$$578x^{2}+39x=123$$

Divide both sides by 578.

$$\frac{578x^{2}+39x}{578}=\frac{123}{578}$$

Dividing by 578 undoes the multiplication by 578.

$$x^{2}+\frac{39}{578}x=\frac{123}{578}$$

Divide \frac{39}{578}, the coefficient of the x term, by 2 to get \frac{39}{1156}. Then add the square of \frac{39}{1156} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{39}{578}x+\left(\frac{39}{1156}\right)^{2}=\frac{123}{578}+\left(\frac{39}{1156}\right)^{2}$$

Square \frac{39}{1156} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{39}{578}x+\frac{1521}{1336336}=\frac{123}{578}+\frac{1521}{1336336}$$

Add \frac{123}{578} to \frac{1521}{1336336} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{39}{578}x+\frac{1521}{1336336}=\frac{285897}{1336336}$$

Factor x^{2}+\frac{39}{578}x+\frac{1521}{1336336}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{39}{1156}\right)^{2}=\frac{285897}{1336336}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{39}{1156}\right)^{2}}=\sqrt{\frac{285897}{1336336}}$$

Simplify.

$$x+\frac{39}{1156}=\frac{\sqrt{285897}}{1156} x+\frac{39}{1156}=-\frac{\sqrt{285897}}{1156}$$

Subtract \frac{39}{1156} from both sides of the equation.

$$x=\frac{\sqrt{285897}-39}{1156} x=\frac{-\sqrt{285897}-39}{1156}$$