Solve 12x=x^2+10x+25

$$12 x = x ^ { 2 } + 10 x + 25$$

Solve for x (complex solution)

$$x=-2\sqrt{6}i+1\approx 1-4.898979486i$$

$$x=1+2\sqrt{6}i\approx 1+4.898979486i$$

Steps Using the Quadratic Formula

Subtract x^{2} from both sides.

$$12x-x^{2}=10x+25$$

Subtract 10x from both sides.

$$12x-x^{2}-10x=25$$

Combine 12x and -10x to get 2x.

$$2x-x^{2}=25$$

Subtract 25 from both sides.

$$2x-x^{2}-25=0$$

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.

$$-x^{2}+2x-25=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 2 for b, and -25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-1\right)\left(-25\right)}}{2\left(-1\right)}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\left(-1\right)\left(-25\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-2±\sqrt{4+4\left(-25\right)}}{2\left(-1\right)}$$

Multiply 4 times -25.

$$x=\frac{-2±\sqrt{4-100}}{2\left(-1\right)}$$

Add 4 to -100.

$$x=\frac{-2±\sqrt{-96}}{2\left(-1\right)}$$

Take the square root of -96.

$$x=\frac{-2±4\sqrt{6}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-2±4\sqrt{6}i}{-2}$$

Now solve the equation x=\frac{-2±4\sqrt{6}i}{-2} when ± is plus. Add -2 to 4i\sqrt{6}.

$$x=\frac{-2+4\sqrt{6}i}{-2}$$

Divide -2+4i\sqrt{6} by -2.

$$x=-2\sqrt{6}i+1$$

Now solve the equation x=\frac{-2±4\sqrt{6}i}{-2} when ± is minus. Subtract 4i\sqrt{6} from -2.

$$x=\frac{-4\sqrt{6}i-2}{-2}$$

Divide -2-4i\sqrt{6} by -2.

$$x=1+2\sqrt{6}i$$

The equation is now solved.

$$x=-2\sqrt{6}i+1 x=1+2\sqrt{6}i$$

Steps for Completing the Square

Subtract x^{2} from both sides.

$$12x-x^{2}=10x+25$$

Subtract 10x from both sides.

$$12x-x^{2}-10x=25$$

Combine 12x and -10x to get 2x.

$$2x-x^{2}=25$$

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.

$$-x^{2}+2x=25$$

Divide both sides by -1.

$$\frac{-x^{2}+2x}{-1}=\frac{25}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{2}{-1}x=\frac{25}{-1}$$

Divide 2 by -1.

$$x^{2}-2x=\frac{25}{-1}$$

Divide 25 by -1.

$$x^{2}-2x=-25$$

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

$$x^{2}-2x+1=-25+1$$

Add -25 to 1.

$$x^{2}-2x+1=-24$$

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

$$\left(x-1\right)^{2}=-24$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{-24}$$

Simplify.

$$x-1=2\sqrt{6}i x-1=-2\sqrt{6}i$$

Add 1 to both sides of the equation.

$$x=1+2\sqrt{6}i x=-2\sqrt{6}i+1$$