Solve 2x^2-x+4=x^2+3x+7

$$2 x ^ { 2 } - x + 4 = x ^ { 2 } + 3 x + 7$$

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.

$$2x^{2}-x+4-x^{2}=3x+7$$

Combine 2x^{2} and -x^{2} to get x^{2}.

$$x^{2}-x+4=3x+7$$

Subtract 3x from both sides.

$$x^{2}-x+4-3x=7$$

Combine -x and -3x to get -4x.

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

Subtract 7 from both sides.

$$x^{2}-4x+4-7=0$$

Subtract 7 from 4 to get -3.

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

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}.

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

Square -4.

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

Multiply -4 times -3.

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

Add 16 to 12.

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

Take the square root of 28.

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

The opposite of -4 is 4.

$$x=\frac{4±2\sqrt{7}}{2}$$

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.

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

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

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

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

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

The equation is now solved.

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

Steps for Completing the Square

Subtract x^{2} from both sides.

$$2x^{2}-x+4-x^{2}=3x+7$$

Combine 2x^{2} and -x^{2} to get x^{2}.

$$x^{2}-x+4=3x+7$$

Subtract 3x from both sides.

$$x^{2}-x+4-3x=7$$

Combine -x and -3x to get -4x.

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

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.

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