Solve 3x^2+36x+48=0

$$3 { x }^{ 2 } +36x+48=0$$

Solve for x

$$x=2\sqrt{5}-6\approx -1.527864045$$

$$x=-2\sqrt{5}-6\approx -10.472135955$$

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.

$$3x^{2}+36x+48=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 36 for b, and 48 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-36±\sqrt{36^{2}-4\times 3\times 48}}{2\times 3}$$

Square 36.

$$x=\frac{-36±\sqrt{1296-4\times 3\times 48}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-36±\sqrt{1296-12\times 48}}{2\times 3}$$

Multiply -12 times 48.

$$x=\frac{-36±\sqrt{1296-576}}{2\times 3}$$

Add 1296 to -576.

$$x=\frac{-36±\sqrt{720}}{2\times 3}$$

Take the square root of 720.

$$x=\frac{-36±12\sqrt{5}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-36±12\sqrt{5}}{6}$$

Now solve the equation x=\frac{-36±12\sqrt{5}}{6} when ± is plus. Add -36 to 12\sqrt{5}.

$$x=\frac{12\sqrt{5}-36}{6}$$

Divide -36+12\sqrt{5} by 6.

$$x=2\sqrt{5}-6$$

Now solve the equation x=\frac{-36±12\sqrt{5}}{6} when ± is minus. Subtract 12\sqrt{5} from -36.

$$x=\frac{-12\sqrt{5}-36}{6}$$

Divide -36-12\sqrt{5} by 6.

$$x=-2\sqrt{5}-6$$

The equation is now solved.

$$x=2\sqrt{5}-6 x=-2\sqrt{5}-6$$

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.

$$3x^{2}+36x+48=0$$

Subtract 48 from both sides of the equation.

$$3x^{2}+36x+48-48=-48$$

Subtracting 48 from itself leaves 0.

$$3x^{2}+36x=-48$$

Divide both sides by 3.

$$\frac{3x^{2}+36x}{3}=-\frac{48}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\frac{36}{3}x=-\frac{48}{3}$$

Divide 36 by 3.

$$x^{2}+12x=-\frac{48}{3}$$

Divide -48 by 3.

$$x^{2}+12x=-16$$

Divide 12, the coefficient of the x term, by 2 to get 6. Then add the square of 6 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+12x+6^{2}=-16+6^{2}$$

Square 6.

$$x^{2}+12x+36=-16+36$$

Add -16 to 36.

$$x^{2}+12x+36=20$$

Factor x^{2}+12x+36. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+6\right)^{2}=20$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+6\right)^{2}}=\sqrt{20}$$

Simplify.

$$x+6=2\sqrt{5} x+6=-2\sqrt{5}$$

Subtract 6 from both sides of the equation.

$$x=2\sqrt{5}-6 x=-2\sqrt{5}-6$$