Solve x^2-3x=2(2-x)

$$x ^ { 2 } - 3 x = 2 ( 2 - x )$$

Solve for x

$$x = \frac{\sqrt{17} + 1}{2} \approx 2.561552813$$

$$x=\frac{1-\sqrt{17}}{2}\approx -1.561552813$$

Steps Using the Quadratic Formula

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

$$x^{2}-3x=4-2x$$

Subtract 4 from both sides.

$$x^{2}-3x-4=-2x$$

Add 2x to both sides.

$$x^{2}-3x-4+2x=0$$

Combine -3x and 2x to get -x.

$$x^{2}-x-4=0$$

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

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

Multiply -4 times -4.

$$x=\frac{-\left(-1\right)±\sqrt{1+16}}{2}$$

Add 1 to 16.

$$x=\frac{-\left(-1\right)±\sqrt{17}}{2}$$

The opposite of -1 is 1.

$$x=\frac{1±\sqrt{17}}{2}$$

Now solve the equation x=\frac{1±\sqrt{17}}{2} when ± is plus. Add 1 to \sqrt{17}.

$$x=\frac{\sqrt{17}+1}{2}$$

Now solve the equation x=\frac{1±\sqrt{17}}{2} when ± is minus. Subtract \sqrt{17} from 1.

$$x=\frac{1-\sqrt{17}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{17}+1}{2} x=\frac{1-\sqrt{17}}{2}$$

Steps for Completing the Square

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

$$x^{2}-3x=4-2x$$

Add 2x to both sides.

$$x^{2}-3x+2x=4$$

Combine -3x and 2x to get -x.

$$x^{2}-x=4$$

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

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

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-x+\frac{1}{4}=4+\frac{1}{4}$$

Add 4 to \frac{1}{4}.

$$x^{2}-x+\frac{1}{4}=\frac{17}{4}$$

Factor x^{2}-x+\frac{1}{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}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{17}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{17}{4}}$$

Simplify.

$$x-\frac{1}{2}=\frac{\sqrt{17}}{2} x-\frac{1}{2}=-\frac{\sqrt{17}}{2}$$

Add \frac{1}{2} to both sides of the equation.

$$x=\frac{\sqrt{17}+1}{2} x=\frac{1-\sqrt{17}}{2}$$