Solve x^2+48x+72=0

$${ x }^{ 2 } +48x+72=0$$

Solve for x

$$x=6\sqrt{14}-24\approx -1.550055679$$

$$x=-6\sqrt{14}-24\approx -46.449944321$$

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.

$$x^{2}+48x+72=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 48 for b, and 72 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-48±\sqrt{48^{2}-4\times 72}}{2}$$

Square 48.

$$x=\frac{-48±\sqrt{2304-4\times 72}}{2}$$

Multiply -4 times 72.

$$x=\frac{-48±\sqrt{2304-288}}{2}$$

Add 2304 to -288.

$$x=\frac{-48±\sqrt{2016}}{2}$$

Take the square root of 2016.

$$x=\frac{-48±12\sqrt{14}}{2}$$

Now solve the equation x=\frac{-48±12\sqrt{14}}{2} when ± is plus. Add -48 to 12\sqrt{14}.

$$x=\frac{12\sqrt{14}-48}{2}$$

Divide -48+12\sqrt{14} by 2.

$$x=6\sqrt{14}-24$$

Now solve the equation x=\frac{-48±12\sqrt{14}}{2} when ± is minus. Subtract 12\sqrt{14} from -48.

$$x=\frac{-12\sqrt{14}-48}{2}$$

Divide -48-12\sqrt{14} by 2.

$$x=-6\sqrt{14}-24$$

The equation is now solved.

$$x=6\sqrt{14}-24 x=-6\sqrt{14}-24$$

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.

$$x^{2}+48x+72=0$$

Subtract 72 from both sides of the equation.

$$x^{2}+48x+72-72=-72$$

Subtracting 72 from itself leaves 0.

$$x^{2}+48x=-72$$

Divide 48, the coefficient of the x term, by 2 to get 24. Then add the square of 24 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+48x+24^{2}=-72+24^{2}$$

Square 24.

$$x^{2}+48x+576=-72+576$$

Add -72 to 576.

$$x^{2}+48x+576=504$$

Factor x^{2}+48x+576. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+24\right)^{2}=504$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+24\right)^{2}}=\sqrt{504}$$

Simplify.

$$x+24=6\sqrt{14} x+24=-6\sqrt{14}$$

Subtract 24 from both sides of the equation.

$$x=6\sqrt{14}-24 x=-6\sqrt{14}-24$$