Solve (10-2x)x=14

$$(10-2x)x=14$$

Solve for x (complex solution)

$$x=\frac{5+\sqrt{3}i}{2}\approx 2.5+0.866025404i$$

$$x=\frac{-\sqrt{3}i+5}{2}\approx 2.5-0.866025404i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 10-2x by x.

$$10x-2x^{2}=14$$

Subtract 14 from both sides.

$$10x-2x^{2}-14=0$$

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.

$$-2x^{2}+10x-14=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 10 for b, and -14 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-10±\sqrt{10^{2}-4\left(-2\right)\left(-14\right)}}{2\left(-2\right)}$$

Square 10.

$$x=\frac{-10±\sqrt{100-4\left(-2\right)\left(-14\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-10±\sqrt{100+8\left(-14\right)}}{2\left(-2\right)}$$

Multiply 8 times -14.

$$x=\frac{-10±\sqrt{100-112}}{2\left(-2\right)}$$

Add 100 to -112.

$$x=\frac{-10±\sqrt{-12}}{2\left(-2\right)}$$

Take the square root of -12.

$$x=\frac{-10±2\sqrt{3}i}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-10±2\sqrt{3}i}{-4}$$

Now solve the equation x=\frac{-10±2\sqrt{3}i}{-4} when ± is plus. Add -10 to 2i\sqrt{3}.

$$x=\frac{-10+2\sqrt{3}i}{-4}$$

Divide -10+2i\sqrt{3} by -4.

$$x=\frac{-\sqrt{3}i+5}{2}$$

Now solve the equation x=\frac{-10±2\sqrt{3}i}{-4} when ± is minus. Subtract 2i\sqrt{3} from -10.

$$x=\frac{-2\sqrt{3}i-10}{-4}$$

Divide -10-2i\sqrt{3} by -4.

$$x=\frac{5+\sqrt{3}i}{2}$$

The equation is now solved.

$$x=\frac{-\sqrt{3}i+5}{2} x=\frac{5+\sqrt{3}i}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 10-2x by x.

$$10x-2x^{2}=14$$

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.

$$-2x^{2}+10x=14$$

Divide both sides by -2.

$$\frac{-2x^{2}+10x}{-2}=\frac{14}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{10}{-2}x=\frac{14}{-2}$$

Divide 10 by -2.

$$x^{2}-5x=\frac{14}{-2}$$

Divide 14 by -2.

$$x^{2}-5x=-7$$

Divide -5, the coefficient of the x term, by 2 to get -\frac{5}{2}. Then add the square of -\frac{5}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-5x+\left(-\frac{5}{2}\right)^{2}=-7+\left(-\frac{5}{2}\right)^{2}$$

Square -\frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-5x+\frac{25}{4}=-7+\frac{25}{4}$$

Add -7 to \frac{25}{4}.

$$x^{2}-5x+\frac{25}{4}=-\frac{3}{4}$$

Factor x^{2}-5x+\frac{25}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{5}{2}\right)^{2}=-\frac{3}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{-\frac{3}{4}}$$

Simplify.

$$x-\frac{5}{2}=\frac{\sqrt{3}i}{2} x-\frac{5}{2}=-\frac{\sqrt{3}i}{2}$$

Add \frac{5}{2} to both sides of the equation.

$$x=\frac{5+\sqrt{3}i}{2} x=\frac{-\sqrt{3}i+5}{2}$$