Solve (x)=x^2-3x+7

$$( x ) = x ^ { 2 } - 3 x + 7$$

Solve for x (complex solution)

$$x=-\sqrt{3}i+2\approx 2-1.732050808i$$

$$x=2+\sqrt{3}i\approx 2+1.732050808i$$

Steps Using the Quadratic Formula

Subtract x^{2} from both sides.

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

Add 3x to both sides.

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

Combine x and 3x to get 4x.

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

Subtract 7 from both sides.

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

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.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 4 for b, and -7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square 4.

$$x=\frac{-4±\sqrt{16-4\left(-1\right)\left(-7\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-4±\sqrt{16+4\left(-7\right)}}{2\left(-1\right)}$$

Multiply 4 times -7.

$$x=\frac{-4±\sqrt{16-28}}{2\left(-1\right)}$$

Add 16 to -28.

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

Take the square root of -12.

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

Multiply 2 times -1.

$$x=\frac{-4±2\sqrt{3}i}{-2}$$

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

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

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

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

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

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

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

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

The equation is now solved.

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

Steps for Completing the Square

Subtract x^{2} from both sides.

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

Add 3x to both sides.

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

Combine x and 3x to get 4x.

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

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.

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

Divide both sides by -1.

$$\frac{-x^{2}+4x}{-1}=\frac{7}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{4}{-1}x=\frac{7}{-1}$$

Divide 4 by -1.

$$x^{2}-4x=\frac{7}{-1}$$

Divide 7 by -1.

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

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.

$$x^{2}-4x+\left(-2\right)^{2}=-7+\left(-2\right)^{2}$$

Square -2.

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

Add -7 to 4.

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

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

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

Add 2 to both sides of the equation.

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