Solve 12x-3=x^2

$$12 x - 3 = x ^ { 2 }$$

Solve for x

$$x=\sqrt{33}+6\approx 11.744562647$$

$$x=6-\sqrt{33}\approx 0.255437353$$

Steps Using the Quadratic Formula

Subtract x^{2} from both sides.

$$12x-3-x^{2}=0$$

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.

$$-x^{2}+12x-3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 12 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\left(-1\right)\left(-3\right)}}{2\left(-1\right)}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\left(-1\right)\left(-3\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-12±\sqrt{144+4\left(-3\right)}}{2\left(-1\right)}$$

Multiply 4 times -3.

$$x=\frac{-12±\sqrt{144-12}}{2\left(-1\right)}$$

Add 144 to -12.

$$x=\frac{-12±\sqrt{132}}{2\left(-1\right)}$$

Take the square root of 132.

$$x=\frac{-12±2\sqrt{33}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-12±2\sqrt{33}}{-2}$$

Now solve the equation x=\frac{-12±2\sqrt{33}}{-2} when ± is plus. Add -12 to 2\sqrt{33}.

$$x=\frac{2\sqrt{33}-12}{-2}$$

Divide -12+2\sqrt{33} by -2.

$$x=6-\sqrt{33}$$

Now solve the equation x=\frac{-12±2\sqrt{33}}{-2} when ± is minus. Subtract 2\sqrt{33} from -12.

$$x=\frac{-2\sqrt{33}-12}{-2}$$

Divide -12-2\sqrt{33} by -2.

$$x=\sqrt{33}+6$$

The equation is now solved.

$$x=6-\sqrt{33} x=\sqrt{33}+6$$

Steps for Completing the Square

Subtract x^{2} from both sides.

$$12x-3-x^{2}=0$$

Add 3 to both sides. Anything plus zero gives itself.

$$12x-x^{2}=3$$

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.

$$-x^{2}+12x=3$$

Divide both sides by -1.

$$\frac{-x^{2}+12x}{-1}=\frac{3}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{12}{-1}x=\frac{3}{-1}$$

Divide 12 by -1.

$$x^{2}-12x=\frac{3}{-1}$$

Divide 3 by -1.

$$x^{2}-12x=-3$$

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.

$$x^{2}-12x+\left(-6\right)^{2}=-3+\left(-6\right)^{2}$$

Square -6.

$$x^{2}-12x+36=-3+36$$

Add -3 to 36.

$$x^{2}-12x+36=33$$

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}.

$$\left(x-6\right)^{2}=33$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-6\right)^{2}}=\sqrt{33}$$

Simplify.

$$x-6=\sqrt{33} x-6=-\sqrt{33}$$

Add 6 to both sides of the equation.

$$x=\sqrt{33}+6 x=6-\sqrt{33}$$