Solve 2x+3=x^2-2x

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

Solve for x

$$x=\sqrt{7}+2\approx 4.645751311$$

$$x=2-\sqrt{7}\approx -0.645751311$$

Steps Using the Quadratic Formula

Subtract x^{2} from both sides.

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

Add 2x to both sides.

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

Combine 2x and 2x to get 4x.

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

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.

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

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

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

Square 4.

$$x=\frac{-4±\sqrt{16-4\left(-1\right)\times 3}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-4±\sqrt{16+4\times 3}}{2\left(-1\right)}$$

Multiply 4 times 3.

$$x=\frac{-4±\sqrt{16+12}}{2\left(-1\right)}$$

Add 16 to 12.

$$x=\frac{-4±\sqrt{28}}{2\left(-1\right)}$$

Take the square root of 28.

$$x=\frac{-4±2\sqrt{7}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-4±2\sqrt{7}}{-2}$$

Now solve the equation x=\frac{-4±2\sqrt{7}}{-2} when ± is plus. Add -4 to 2\sqrt{7}.

$$x=\frac{2\sqrt{7}-4}{-2}$$

Divide -4+2\sqrt{7} by -2.

$$x=2-\sqrt{7}$$

Now solve the equation x=\frac{-4±2\sqrt{7}}{-2} when ± is minus. Subtract 2\sqrt{7} from -4.

$$x=\frac{-2\sqrt{7}-4}{-2}$$

Divide -4-2\sqrt{7} by -2.

$$x=\sqrt{7}+2$$

The equation is now solved.

$$x=2-\sqrt{7} x=\sqrt{7}+2$$

Steps for Completing the Square

Subtract x^{2} from both sides.

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

Add 2x to both sides.

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

Combine 2x and 2x to get 4x.

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

Subtract 3 from both sides. Anything subtracted from zero gives its negation.

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

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.

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

Divide both sides by -1.

$$\frac{-x^{2}+4x}{-1}=-\frac{3}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{4}{-1}x=-\frac{3}{-1}$$

Divide 4 by -1.

$$x^{2}-4x=-\frac{3}{-1}$$

Divide -3 by -1.

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

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

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

Square -2.

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

Add 3 to 4.

$$x^{2}-4x+4=7$$

Factor x^{2}-4x+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}.

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

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{7}$$

Simplify.

$$x-2=\sqrt{7} x-2=-\sqrt{7}$$

Add 2 to both sides of the equation.

$$x=\sqrt{7}+2 x=2-\sqrt{7}$$