Solve x^2+28x-2000=0

$${ x }^{ 2 } +28x-2000=0$$

Solve for x

$$x=6\sqrt{61}-14\approx 32.861498055$$

$$x=-6\sqrt{61}-14\approx -60.861498055$$

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.

$$x^{2}+28x-2000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 28 for b, and -2000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-28±\sqrt{28^{2}-4\left(-2000\right)}}{2}$$

Square 28.

$$x=\frac{-28±\sqrt{784-4\left(-2000\right)}}{2}$$

Multiply -4 times -2000.

$$x=\frac{-28±\sqrt{784+8000}}{2}$$

Add 784 to 8000.

$$x=\frac{-28±\sqrt{8784}}{2}$$

Take the square root of 8784.

$$x=\frac{-28±12\sqrt{61}}{2}$$

Now solve the equation x=\frac{-28±12\sqrt{61}}{2} when ± is plus. Add -28 to 12\sqrt{61}.

$$x=\frac{12\sqrt{61}-28}{2}$$

Divide -28+12\sqrt{61} by 2.

$$x=6\sqrt{61}-14$$

Now solve the equation x=\frac{-28±12\sqrt{61}}{2} when ± is minus. Subtract 12\sqrt{61} from -28.

$$x=\frac{-12\sqrt{61}-28}{2}$$

Divide -28-12\sqrt{61} by 2.

$$x=-6\sqrt{61}-14$$

The equation is now solved.

$$x=6\sqrt{61}-14 x=-6\sqrt{61}-14$$

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.

$$x^{2}+28x-2000=0$$

Add 2000 to both sides of the equation.

$$x^{2}+28x-2000-\left(-2000\right)=-\left(-2000\right)$$

Subtracting -2000 from itself leaves 0.

$$x^{2}+28x=-\left(-2000\right)$$

Subtract -2000 from 0.

$$x^{2}+28x=2000$$

Divide 28, the coefficient of the x term, by 2 to get 14. Then add the square of 14 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+28x+14^{2}=2000+14^{2}$$

Square 14.

$$x^{2}+28x+196=2000+196$$

Add 2000 to 196.

$$x^{2}+28x+196=2196$$

Factor x^{2}+28x+196. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+14\right)^{2}=2196$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+14\right)^{2}}=\sqrt{2196}$$

Simplify.

$$x+14=6\sqrt{61} x+14=-6\sqrt{61}$$

Subtract 14 from both sides of the equation.

$$x=6\sqrt{61}-14 x=-6\sqrt{61}-14$$