Solve (12-2x+1)x=80

$$(12-2x+1)x=80$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{471}i+13}{4}\approx 3.25-5.425633604i$$

$$x=\frac{13+\sqrt{471}i}{4}\approx 3.25+5.425633604i$$

Steps Using the Quadratic Formula

Add 12 and 1 to get 13.

$$\left(13-2x\right)x=80$$

Use the distributive property to multiply 13-2x by x.

$$13x-2x^{2}=80$$

Subtract 80 from both sides.

$$13x-2x^{2}-80=0$$

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.

$$-2x^{2}+13x-80=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 13 for b, and -80 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-13±\sqrt{13^{2}-4\left(-2\right)\left(-80\right)}}{2\left(-2\right)}$$

Square 13.

$$x=\frac{-13±\sqrt{169-4\left(-2\right)\left(-80\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-13±\sqrt{169+8\left(-80\right)}}{2\left(-2\right)}$$

Multiply 8 times -80.

$$x=\frac{-13±\sqrt{169-640}}{2\left(-2\right)}$$

Add 169 to -640.

$$x=\frac{-13±\sqrt{-471}}{2\left(-2\right)}$$

Take the square root of -471.

$$x=\frac{-13±\sqrt{471}i}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-13±\sqrt{471}i}{-4}$$

Now solve the equation x=\frac{-13±\sqrt{471}i}{-4} when ± is plus. Add -13 to i\sqrt{471}.

$$x=\frac{-13+\sqrt{471}i}{-4}$$

Divide -13+i\sqrt{471} by -4.

$$x=\frac{-\sqrt{471}i+13}{4}$$

Now solve the equation x=\frac{-13±\sqrt{471}i}{-4} when ± is minus. Subtract i\sqrt{471} from -13.

$$x=\frac{-\sqrt{471}i-13}{-4}$$

Divide -13-i\sqrt{471} by -4.

$$x=\frac{13+\sqrt{471}i}{4}$$

The equation is now solved.

$$x=\frac{-\sqrt{471}i+13}{4} x=\frac{13+\sqrt{471}i}{4}$$

Steps for Completing the Square

Add 12 and 1 to get 13.

$$\left(13-2x\right)x=80$$

Use the distributive property to multiply 13-2x by x.

$$13x-2x^{2}=80$$

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.

$$-2x^{2}+13x=80$$

Divide both sides by -2.

$$\frac{-2x^{2}+13x}{-2}=\frac{80}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{13}{-2}x=\frac{80}{-2}$$

Divide 13 by -2.

$$x^{2}-\frac{13}{2}x=\frac{80}{-2}$$

Divide 80 by -2.

$$x^{2}-\frac{13}{2}x=-40$$

Divide -\frac{13}{2}, the coefficient of the x term, by 2 to get -\frac{13}{4}. Then add the square of -\frac{13}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{13}{2}x+\left(-\frac{13}{4}\right)^{2}=-40+\left(-\frac{13}{4}\right)^{2}$$

Square -\frac{13}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{13}{2}x+\frac{169}{16}=-40+\frac{169}{16}$$

Add -40 to \frac{169}{16}.

$$x^{2}-\frac{13}{2}x+\frac{169}{16}=-\frac{471}{16}$$

Factor x^{2}-\frac{13}{2}x+\frac{169}{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}.

$$\left(x-\frac{13}{4}\right)^{2}=-\frac{471}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{13}{4}\right)^{2}}=\sqrt{-\frac{471}{16}}$$

Simplify.

$$x-\frac{13}{4}=\frac{\sqrt{471}i}{4} x-\frac{13}{4}=-\frac{\sqrt{471}i}{4}$$

Add \frac{13}{4} to both sides of the equation.

$$x=\frac{13+\sqrt{471}i}{4} x=\frac{-\sqrt{471}i+13}{4}$$