Solve 90x^2+8x-400=60

$$90 x ^ { 2 } + 8 x - 400 = 60$$

Solve for x

$$x = \frac{\sqrt{10354} - 2}{45} \approx 2.216769039$$

$$x=\frac{-\sqrt{10354}-2}{45}\approx -2.305657928$$

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.

$$90x^{2}+8x-400=60$$

Subtract 60 from both sides of the equation.

$$90x^{2}+8x-400-60=60-60$$

Subtracting 60 from itself leaves 0.

$$90x^{2}+8x-400-60=0$$

Subtract 60 from -400.

$$90x^{2}+8x-460=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 90 for a, 8 for b, and -460 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\times 90\left(-460\right)}}{2\times 90}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\times 90\left(-460\right)}}{2\times 90}$$

Multiply -4 times 90.

$$x=\frac{-8±\sqrt{64-360\left(-460\right)}}{2\times 90}$$

Multiply -360 times -460.

$$x=\frac{-8±\sqrt{64+165600}}{2\times 90}$$

Add 64 to 165600.

$$x=\frac{-8±\sqrt{165664}}{2\times 90}$$

Take the square root of 165664.

$$x=\frac{-8±4\sqrt{10354}}{2\times 90}$$

Multiply 2 times 90.

$$x=\frac{-8±4\sqrt{10354}}{180}$$

Now solve the equation x=\frac{-8±4\sqrt{10354}}{180} when ± is plus. Add -8 to 4\sqrt{10354}.

$$x=\frac{4\sqrt{10354}-8}{180}$$

Divide -8+4\sqrt{10354} by 180.

$$x=\frac{\sqrt{10354}-2}{45}$$

Now solve the equation x=\frac{-8±4\sqrt{10354}}{180} when ± is minus. Subtract 4\sqrt{10354} from -8.

$$x=\frac{-4\sqrt{10354}-8}{180}$$

Divide -8-4\sqrt{10354} by 180.

$$x=\frac{-\sqrt{10354}-2}{45}$$

The equation is now solved.

$$x=\frac{\sqrt{10354}-2}{45} x=\frac{-\sqrt{10354}-2}{45}$$

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.

$$90x^{2}+8x-400=60$$

Add 400 to both sides of the equation.

$$90x^{2}+8x-400-\left(-400\right)=60-\left(-400\right)$$

Subtracting -400 from itself leaves 0.

$$90x^{2}+8x=60-\left(-400\right)$$

Subtract -400 from 60.

$$90x^{2}+8x=460$$

Divide both sides by 90.

$$\frac{90x^{2}+8x}{90}=\frac{460}{90}$$

Dividing by 90 undoes the multiplication by 90.

$$x^{2}+\frac{8}{90}x=\frac{460}{90}$$

Reduce the fraction \frac{8}{90} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{4}{45}x=\frac{460}{90}$$

Reduce the fraction \frac{460}{90} to lowest terms by extracting and canceling out 10.

$$x^{2}+\frac{4}{45}x=\frac{46}{9}$$

Divide \frac{4}{45}, the coefficient of the x term, by 2 to get \frac{2}{45}. Then add the square of \frac{2}{45} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{4}{45}x+\left(\frac{2}{45}\right)^{2}=\frac{46}{9}+\left(\frac{2}{45}\right)^{2}$$

Square \frac{2}{45} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{4}{45}x+\frac{4}{2025}=\frac{46}{9}+\frac{4}{2025}$$

Add \frac{46}{9} to \frac{4}{2025} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{4}{45}x+\frac{4}{2025}=\frac{10354}{2025}$$

Factor x^{2}+\frac{4}{45}x+\frac{4}{2025}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{2}{45}\right)^{2}=\frac{10354}{2025}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{2}{45}\right)^{2}}=\sqrt{\frac{10354}{2025}}$$

Simplify.

$$x+\frac{2}{45}=\frac{\sqrt{10354}}{45} x+\frac{2}{45}=-\frac{\sqrt{10354}}{45}$$

Subtract \frac{2}{45} from both sides of the equation.

$$x=\frac{\sqrt{10354}-2}{45} x=\frac{-\sqrt{10354}-2}{45}$$