Solve 4x-15=(x)^2+13

$$4 x - 15 = ( x ) ^ { 2 } + 13$$

Solve for x (complex solution)

$$x=-2\sqrt{6}i+2\approx 2-4.898979486i$$

$$x=2+2\sqrt{6}i\approx 2+4.898979486i$$

Steps Using the Quadratic Formula

Subtract x^{2} from both sides.

$$4x-15-x^{2}=13$$

Subtract 13 from both sides.

$$4x-15-x^{2}-13=0$$

Subtract 13 from -15 to get -28.

$$4x-28-x^{2}=0$$

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.

$$-x^{2}+4x-28=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 4 for b, and -28 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square 4.

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

Multiply -4 times -1.

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

Multiply 4 times -28.

$$x=\frac{-4±\sqrt{16-112}}{2\left(-1\right)}$$

Add 16 to -112.

$$x=\frac{-4±\sqrt{-96}}{2\left(-1\right)}$$

Take the square root of -96.

$$x=\frac{-4±4\sqrt{6}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-4±4\sqrt{6}i}{-2}$$

Now solve the equation x=\frac{-4±4\sqrt{6}i}{-2} when ± is plus. Add -4 to 4i\sqrt{6}.

$$x=\frac{-4+4\sqrt{6}i}{-2}$$

Divide -4+4i\sqrt{6} by -2.

$$x=-2\sqrt{6}i+2$$

Now solve the equation x=\frac{-4±4\sqrt{6}i}{-2} when ± is minus. Subtract 4i\sqrt{6} from -4.

$$x=\frac{-4\sqrt{6}i-4}{-2}$$

Divide -4-4i\sqrt{6} by -2.

$$x=2+2\sqrt{6}i$$

The equation is now solved.

$$x=-2\sqrt{6}i+2 x=2+2\sqrt{6}i$$

Steps for Completing the Square

Subtract x^{2} from both sides.

$$4x-15-x^{2}=13$$

Add 15 to both sides.

$$4x-x^{2}=13+15$$

Add 13 and 15 to get 28.

$$4x-x^{2}=28$$

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.

$$-x^{2}+4x=28$$

Divide both sides by -1.

$$\frac{-x^{2}+4x}{-1}=\frac{28}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{4}{-1}x=\frac{28}{-1}$$

Divide 4 by -1.

$$x^{2}-4x=\frac{28}{-1}$$

Divide 28 by -1.

$$x^{2}-4x=-28$$

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.

$$x^{2}-4x+\left(-2\right)^{2}=-28+\left(-2\right)^{2}$$

Square -2.

$$x^{2}-4x+4=-28+4$$

Add -28 to 4.

$$x^{2}-4x+4=-24$$

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}.

$$\left(x-2\right)^{2}=-24$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{-24}$$

Simplify.

$$x-2=2\sqrt{6}i x-2=-2\sqrt{6}i$$

Add 2 to both sides of the equation.

$$x=2+2\sqrt{6}i x=-2\sqrt{6}i+2$$