Solve x^2+90*x-6500=0

$$x ^ { 2 } + 90 \cdot x - 6500 = 0$$

Solve for x

$$x=5\sqrt{341}-45\approx 47.330926563$$

$$x=-5\sqrt{341}-45\approx -137.330926563$$

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.

$$x^{2}+90x-6500=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 90 for b, and -6500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-90±\sqrt{90^{2}-4\left(-6500\right)}}{2}$$

Square 90.

$$x=\frac{-90±\sqrt{8100-4\left(-6500\right)}}{2}$$

Multiply -4 times -6500.

$$x=\frac{-90±\sqrt{8100+26000}}{2}$$

Add 8100 to 26000.

$$x=\frac{-90±\sqrt{34100}}{2}$$

Take the square root of 34100.

$$x=\frac{-90±10\sqrt{341}}{2}$$

Now solve the equation x=\frac{-90±10\sqrt{341}}{2} when ± is plus. Add -90 to 10\sqrt{341}.

$$x=\frac{10\sqrt{341}-90}{2}$$

Divide -90+10\sqrt{341} by 2.

$$x=5\sqrt{341}-45$$

Now solve the equation x=\frac{-90±10\sqrt{341}}{2} when ± is minus. Subtract 10\sqrt{341} from -90.

$$x=\frac{-10\sqrt{341}-90}{2}$$

Divide -90-10\sqrt{341} by 2.

$$x=-5\sqrt{341}-45$$

The equation is now solved.

$$x=5\sqrt{341}-45 x=-5\sqrt{341}-45$$

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.

$$x^{2}+90x-6500=0$$

Add 6500 to both sides of the equation.

$$x^{2}+90x-6500-\left(-6500\right)=-\left(-6500\right)$$

Subtracting -6500 from itself leaves 0.

$$x^{2}+90x=-\left(-6500\right)$$

Subtract -6500 from 0.

$$x^{2}+90x=6500$$

Divide 90, the coefficient of the x term, by 2 to get 45. Then add the square of 45 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+90x+45^{2}=6500+45^{2}$$

Square 45.

$$x^{2}+90x+2025=6500+2025$$

Add 6500 to 2025.

$$x^{2}+90x+2025=8525$$

Factor x^{2}+90x+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}.

$$\left(x+45\right)^{2}=8525$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+45\right)^{2}}=\sqrt{8525}$$

Simplify.

$$x+45=5\sqrt{341} x+45=-5\sqrt{341}$$

Subtract 45 from both sides of the equation.

$$x=5\sqrt{341}-45 x=-5\sqrt{341}-45$$