Solve 24x^2+46x+24=0

$$24 { x }^{ 2 } +46x+24=0$$

Solve for x (complex solution)

$$x=\frac{-23+\sqrt{47}i}{24}\approx -0.958333333+0.285652275i$$

$$x=\frac{-\sqrt{47}i-23}{24}\approx -0.958333333-0.285652275i$$

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.

$$24x^{2}+46x+24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 24 for a, 46 for b, and 24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-46±\sqrt{46^{2}-4\times 24\times 24}}{2\times 24}$$

Square 46.

$$x=\frac{-46±\sqrt{2116-4\times 24\times 24}}{2\times 24}$$

Multiply -4 times 24.

$$x=\frac{-46±\sqrt{2116-96\times 24}}{2\times 24}$$

Multiply -96 times 24.

$$x=\frac{-46±\sqrt{2116-2304}}{2\times 24}$$

Add 2116 to -2304.

$$x=\frac{-46±\sqrt{-188}}{2\times 24}$$

Take the square root of -188.

$$x=\frac{-46±2\sqrt{47}i}{2\times 24}$$

Multiply 2 times 24.

$$x=\frac{-46±2\sqrt{47}i}{48}$$

Now solve the equation x=\frac{-46±2\sqrt{47}i}{48} when ± is plus. Add -46 to 2i\sqrt{47}.

$$x=\frac{-46+2\sqrt{47}i}{48}$$

Divide -46+2i\sqrt{47} by 48.

$$x=\frac{-23+\sqrt{47}i}{24}$$

Now solve the equation x=\frac{-46±2\sqrt{47}i}{48} when ± is minus. Subtract 2i\sqrt{47} from -46.

$$x=\frac{-2\sqrt{47}i-46}{48}$$

Divide -46-2i\sqrt{47} by 48.

$$x=\frac{-\sqrt{47}i-23}{24}$$

The equation is now solved.

$$x=\frac{-23+\sqrt{47}i}{24} x=\frac{-\sqrt{47}i-23}{24}$$

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.

$$24x^{2}+46x+24=0$$

Subtract 24 from both sides of the equation.

$$24x^{2}+46x+24-24=-24$$

Subtracting 24 from itself leaves 0.

$$24x^{2}+46x=-24$$

Divide both sides by 24.

$$\frac{24x^{2}+46x}{24}=-\frac{24}{24}$$

Dividing by 24 undoes the multiplication by 24.

$$x^{2}+\frac{46}{24}x=-\frac{24}{24}$$

Reduce the fraction \frac{46}{24} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{23}{12}x=-\frac{24}{24}$$

Divide -24 by 24.

$$x^{2}+\frac{23}{12}x=-1$$

Divide \frac{23}{12}, the coefficient of the x term, by 2 to get \frac{23}{24}. Then add the square of \frac{23}{24} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{23}{12}x+\left(\frac{23}{24}\right)^{2}=-1+\left(\frac{23}{24}\right)^{2}$$

Square \frac{23}{24} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{23}{12}x+\frac{529}{576}=-1+\frac{529}{576}$$

Add -1 to \frac{529}{576}.

$$x^{2}+\frac{23}{12}x+\frac{529}{576}=-\frac{47}{576}$$

Factor x^{2}+\frac{23}{12}x+\frac{529}{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}.

$$\left(x+\frac{23}{24}\right)^{2}=-\frac{47}{576}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{23}{24}\right)^{2}}=\sqrt{-\frac{47}{576}}$$

Simplify.

$$x+\frac{23}{24}=\frac{\sqrt{47}i}{24} x+\frac{23}{24}=-\frac{\sqrt{47}i}{24}$$

Subtract \frac{23}{24} from both sides of the equation.

$$x=\frac{-23+\sqrt{47}i}{24} x=\frac{-\sqrt{47}i-23}{24}$$