Solve 3x^2*16-24x+24=0

$$3 { x }^{ 2 } \times 16-24x+24=0$$

Solve for x (complex solution)

$$x=\frac{1+\sqrt{7}i}{4}\approx 0.25+0.661437828i$$

$$x=\frac{-\sqrt{7}i+1}{4}\approx 0.25-0.661437828i$$

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.

$$48x^{2}-24x+24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 48 for a, -24 for b, and 24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 48\times 24}}{2\times 48}$$

Square -24.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\times 48\times 24}}{2\times 48}$$

Multiply -4 times 48.

$$x=\frac{-\left(-24\right)±\sqrt{576-192\times 24}}{2\times 48}$$

Multiply -192 times 24.

$$x=\frac{-\left(-24\right)±\sqrt{576-4608}}{2\times 48}$$

Add 576 to -4608.

$$x=\frac{-\left(-24\right)±\sqrt{-4032}}{2\times 48}$$

Take the square root of -4032.

$$x=\frac{-\left(-24\right)±24\sqrt{7}i}{2\times 48}$$

The opposite of -24 is 24.

$$x=\frac{24±24\sqrt{7}i}{2\times 48}$$

Multiply 2 times 48.

$$x=\frac{24±24\sqrt{7}i}{96}$$

Now solve the equation x=\frac{24±24\sqrt{7}i}{96} when ± is plus. Add 24 to 24i\sqrt{7}.

$$x=\frac{24+24\sqrt{7}i}{96}$$

Divide 24+24i\sqrt{7} by 96.

$$x=\frac{1+\sqrt{7}i}{4}$$

Now solve the equation x=\frac{24±24\sqrt{7}i}{96} when ± is minus. Subtract 24i\sqrt{7} from 24.

$$x=\frac{-24\sqrt{7}i+24}{96}$$

Divide 24-24i\sqrt{7} by 96.

$$x=\frac{-\sqrt{7}i+1}{4}$$

The equation is now solved.

$$x=\frac{1+\sqrt{7}i}{4} x=\frac{-\sqrt{7}i+1}{4}$$

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.

$$48x^{2}-24x+24=0$$

Subtract 24 from both sides of the equation.

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

Subtracting 24 from itself leaves 0.

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

Divide both sides by 48.

$$\frac{48x^{2}-24x}{48}=-\frac{24}{48}$$

Dividing by 48 undoes the multiplication by 48.

$$x^{2}+\left(-\frac{24}{48}\right)x=-\frac{24}{48}$$

Reduce the fraction \frac{-24}{48} to lowest terms by extracting and canceling out 24.

$$x^{2}-\frac{1}{2}x=-\frac{24}{48}$$

Reduce the fraction \frac{-24}{48} to lowest terms by extracting and canceling out 24.

$$x^{2}-\frac{1}{2}x=-\frac{1}{2}$$

Divide -\frac{1}{2}, the coefficient of the x term, by 2 to get -\frac{1}{4}. Then add the square of -\frac{1}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1}{2}x+\left(-\frac{1}{4}\right)^{2}=-\frac{1}{2}+\left(-\frac{1}{4}\right)^{2}$$

Square -\frac{1}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=-\frac{1}{2}+\frac{1}{16}$$

Add -\frac{1}{2} to \frac{1}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=-\frac{7}{16}$$

Factor x^{2}-\frac{1}{2}x+\frac{1}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{4}\right)^{2}=-\frac{7}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{-\frac{7}{16}}$$

Simplify.

$$x-\frac{1}{4}=\frac{\sqrt{7}i}{4} x-\frac{1}{4}=-\frac{\sqrt{7}i}{4}$$

Add \frac{1}{4} to both sides of the equation.

$$x=\frac{1+\sqrt{7}i}{4} x=\frac{-\sqrt{7}i+1}{4}$$