Solve 2*x+5=x^2+9

$$2 \cdot x + 5 = x ^ { 2 } + 9$$

Solve for x (complex solution)

$$x=-\sqrt{3}i+1\approx 1-1.732050808i$$

$$x=1+\sqrt{3}i\approx 1+1.732050808i$$

Steps Using the Quadratic Formula

Subtract x^{2} from both sides.

$$2x+5-x^{2}=9$$

Subtract 9 from both sides.

$$2x+5-x^{2}-9=0$$

Subtract 9 from 5 to get -4.

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

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.

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

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

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

Square 2.

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

Multiply -4 times -1.

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

Multiply 4 times -4.

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

Add 4 to -16.

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

Take the square root of -12.

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

Multiply 2 times -1.

$$x=\frac{-2±2\sqrt{3}i}{-2}$$

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

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

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

$$x=-\sqrt{3}i+1$$

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

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

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

$$x=1+\sqrt{3}i$$

The equation is now solved.

$$x=-\sqrt{3}i+1 x=1+\sqrt{3}i$$

Steps for Completing the Square

Subtract x^{2} from both sides.

$$2x+5-x^{2}=9$$

Subtract 5 from both sides.

$$2x-x^{2}=9-5$$

Subtract 5 from 9 to get 4.

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

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.

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

Divide both sides by -1.

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

Dividing by -1 undoes the multiplication by -1.

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

Divide 2 by -1.

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

Divide 4 by -1.

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

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

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

Add -4 to 1.

$$x^{2}-2x+1=-3$$

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1\right)^{2}=-3$$

Take the square root of both sides of the equation.

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

Simplify.

$$x-1=\sqrt{3}i x-1=-\sqrt{3}i$$

Add 1 to both sides of the equation.

$$x=1+\sqrt{3}i x=-\sqrt{3}i+1$$