Solve 48-12x^2-49*3=32x

$$48-12 { x }^{ 2 } -49 \times 3=32x$$

Solve for x (complex solution)

$$x=-\frac{\sqrt{233}i}{6}-\frac{4}{3}\approx -1.333333333-2.544056254i$$

$$x=\frac{\sqrt{233}i}{6}-\frac{4}{3}\approx -1.333333333+2.544056254i$$

Steps Using the Quadratic Formula

Multiply 49 and 3 to get 147.

$$48-12x^{2}-147=32x$$

Subtract 147 from 48 to get -99.

$$-99-12x^{2}=32x$$

Subtract 32x from both sides.

$$-99-12x^{2}-32x=0$$

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.

$$-12x^{2}-32x-99=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -12 for a, -32 for b, and -99 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-32\right)±\sqrt{\left(-32\right)^{2}-4\left(-12\right)\left(-99\right)}}{2\left(-12\right)}$$

Square -32.

$$x=\frac{-\left(-32\right)±\sqrt{1024-4\left(-12\right)\left(-99\right)}}{2\left(-12\right)}$$

Multiply -4 times -12.

$$x=\frac{-\left(-32\right)±\sqrt{1024+48\left(-99\right)}}{2\left(-12\right)}$$

Multiply 48 times -99.

$$x=\frac{-\left(-32\right)±\sqrt{1024-4752}}{2\left(-12\right)}$$

Add 1024 to -4752.

$$x=\frac{-\left(-32\right)±\sqrt{-3728}}{2\left(-12\right)}$$

Take the square root of -3728.

$$x=\frac{-\left(-32\right)±4\sqrt{233}i}{2\left(-12\right)}$$

The opposite of -32 is 32.

$$x=\frac{32±4\sqrt{233}i}{2\left(-12\right)}$$

Multiply 2 times -12.

$$x=\frac{32±4\sqrt{233}i}{-24}$$

Now solve the equation x=\frac{32±4\sqrt{233}i}{-24} when ± is plus. Add 32 to 4i\sqrt{233}.

$$x=\frac{32+4\sqrt{233}i}{-24}$$

Divide 32+4i\sqrt{233} by -24.

$$x=-\frac{\sqrt{233}i}{6}-\frac{4}{3}$$

Now solve the equation x=\frac{32±4\sqrt{233}i}{-24} when ± is minus. Subtract 4i\sqrt{233} from 32.

$$x=\frac{-4\sqrt{233}i+32}{-24}$$

Divide 32-4i\sqrt{233} by -24.

$$x=\frac{\sqrt{233}i}{6}-\frac{4}{3}$$

The equation is now solved.

$$x=-\frac{\sqrt{233}i}{6}-\frac{4}{3} x=\frac{\sqrt{233}i}{6}-\frac{4}{3}$$

Steps for Completing the Square

Multiply 49 and 3 to get 147.

$$48-12x^{2}-147=32x$$

Subtract 147 from 48 to get -99.

$$-99-12x^{2}=32x$$

Subtract 32x from both sides.

$$-99-12x^{2}-32x=0$$

Add 99 to both sides. Anything plus zero gives itself.

$$-12x^{2}-32x=99$$

Divide both sides by -12.

$$\frac{-12x^{2}-32x}{-12}=\frac{99}{-12}$$

Dividing by -12 undoes the multiplication by -12.

$$x^{2}+\left(-\frac{32}{-12}\right)x=\frac{99}{-12}$$

Reduce the fraction \frac{-32}{-12} to lowest terms by extracting and canceling out 4.

$$x^{2}+\frac{8}{3}x=\frac{99}{-12}$$

Reduce the fraction \frac{99}{-12} to lowest terms by extracting and canceling out 3.

$$x^{2}+\frac{8}{3}x=-\frac{33}{4}$$

Divide \frac{8}{3}, the coefficient of the x term, by 2 to get \frac{4}{3}. Then add the square of \frac{4}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{8}{3}x+\left(\frac{4}{3}\right)^{2}=-\frac{33}{4}+\left(\frac{4}{3}\right)^{2}$$

Square \frac{4}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{8}{3}x+\frac{16}{9}=-\frac{33}{4}+\frac{16}{9}$$

Add -\frac{33}{4} to \frac{16}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{8}{3}x+\frac{16}{9}=-\frac{233}{36}$$

Factor x^{2}+\frac{8}{3}x+\frac{16}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{4}{3}\right)^{2}=-\frac{233}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{4}{3}\right)^{2}}=\sqrt{-\frac{233}{36}}$$

Simplify.

$$x+\frac{4}{3}=\frac{\sqrt{233}i}{6} x+\frac{4}{3}=-\frac{\sqrt{233}i}{6}$$

Subtract \frac{4}{3} from both sides of the equation.

$$x=\frac{\sqrt{233}i}{6}-\frac{4}{3} x=-\frac{\sqrt{233}i}{6}-\frac{4}{3}$$