Solve x^2+60x-1050=0

$${ x }^{ 2 } +60x-1050=0$$

Solve for x

$$x=5\sqrt{78}-30\approx 14.158804332$$

$$x=-5\sqrt{78}-30\approx -74.158804332$$

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.

$$x^{2}+60x-1050=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 60 for b, and -1050 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-60±\sqrt{60^{2}-4\left(-1050\right)}}{2}$$

Square 60.

$$x=\frac{-60±\sqrt{3600-4\left(-1050\right)}}{2}$$

Multiply -4 times -1050.

$$x=\frac{-60±\sqrt{3600+4200}}{2}$$

Add 3600 to 4200.

$$x=\frac{-60±\sqrt{7800}}{2}$$

Take the square root of 7800.

$$x=\frac{-60±10\sqrt{78}}{2}$$

Now solve the equation x=\frac{-60±10\sqrt{78}}{2} when ± is plus. Add -60 to 10\sqrt{78}.

$$x=\frac{10\sqrt{78}-60}{2}$$

Divide -60+10\sqrt{78} by 2.

$$x=5\sqrt{78}-30$$

Now solve the equation x=\frac{-60±10\sqrt{78}}{2} when ± is minus. Subtract 10\sqrt{78} from -60.

$$x=\frac{-10\sqrt{78}-60}{2}$$

Divide -60-10\sqrt{78} by 2.

$$x=-5\sqrt{78}-30$$

The equation is now solved.

$$x=5\sqrt{78}-30 x=-5\sqrt{78}-30$$

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.

$$x^{2}+60x-1050=0$$

Add 1050 to both sides of the equation.

$$x^{2}+60x-1050-\left(-1050\right)=-\left(-1050\right)$$

Subtracting -1050 from itself leaves 0.

$$x^{2}+60x=-\left(-1050\right)$$

Subtract -1050 from 0.

$$x^{2}+60x=1050$$

Divide 60, the coefficient of the x term, by 2 to get 30. Then add the square of 30 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+60x+30^{2}=1050+30^{2}$$

Square 30.

$$x^{2}+60x+900=1050+900$$

Add 1050 to 900.

$$x^{2}+60x+900=1950$$

Factor x^{2}+60x+900. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+30\right)^{2}=1950$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+30\right)^{2}}=\sqrt{1950}$$

Simplify.

$$x+30=5\sqrt{78} x+30=-5\sqrt{78}$$

Subtract 30 from both sides of the equation.

$$x=5\sqrt{78}-30 x=-5\sqrt{78}-30$$