Solve 40x+15=8x^2+24x+78

$$40 x + 15 = 8 x ^ { 2 } + 24 x + 78$$

Solve for x (complex solution)

$$x=-\frac{\sqrt{110}i}{4}+1\approx 1-2.62202212i$$

$$x=\frac{\sqrt{110}i}{4}+1\approx 1+2.62202212i$$

Steps Using the Quadratic Formula

Subtract 8x^{2} from both sides.

$$40x+15-8x^{2}=24x+78$$

Subtract 24x from both sides.

$$40x+15-8x^{2}-24x=78$$

Combine 40x and -24x to get 16x.

$$16x+15-8x^{2}=78$$

Subtract 78 from both sides.

$$16x+15-8x^{2}-78=0$$

Subtract 78 from 15 to get -63.

$$16x-63-8x^{2}=0$$

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.

$$-8x^{2}+16x-63=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -8 for a, 16 for b, and -63 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\left(-8\right)\left(-63\right)}}{2\left(-8\right)}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\left(-8\right)\left(-63\right)}}{2\left(-8\right)}$$

Multiply -4 times -8.

$$x=\frac{-16±\sqrt{256+32\left(-63\right)}}{2\left(-8\right)}$$

Multiply 32 times -63.

$$x=\frac{-16±\sqrt{256-2016}}{2\left(-8\right)}$$

Add 256 to -2016.

$$x=\frac{-16±\sqrt{-1760}}{2\left(-8\right)}$$

Take the square root of -1760.

$$x=\frac{-16±4\sqrt{110}i}{2\left(-8\right)}$$

Multiply 2 times -8.

$$x=\frac{-16±4\sqrt{110}i}{-16}$$

Now solve the equation x=\frac{-16±4\sqrt{110}i}{-16} when ± is plus. Add -16 to 4i\sqrt{110}.

$$x=\frac{-16+4\sqrt{110}i}{-16}$$

Divide -16+4i\sqrt{110} by -16.

$$x=-\frac{\sqrt{110}i}{4}+1$$

Now solve the equation x=\frac{-16±4\sqrt{110}i}{-16} when ± is minus. Subtract 4i\sqrt{110} from -16.

$$x=\frac{-4\sqrt{110}i-16}{-16}$$

Divide -16-4i\sqrt{110} by -16.

$$x=\frac{\sqrt{110}i}{4}+1$$

The equation is now solved.

$$x=-\frac{\sqrt{110}i}{4}+1 x=\frac{\sqrt{110}i}{4}+1$$

Steps for Completing the Square

Subtract 8x^{2} from both sides.

$$40x+15-8x^{2}=24x+78$$

Subtract 24x from both sides.

$$40x+15-8x^{2}-24x=78$$

Combine 40x and -24x to get 16x.

$$16x+15-8x^{2}=78$$

Subtract 15 from both sides.

$$16x-8x^{2}=78-15$$

Subtract 15 from 78 to get 63.

$$16x-8x^{2}=63$$

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.

$$-8x^{2}+16x=63$$

Divide both sides by -8.

$$\frac{-8x^{2}+16x}{-8}=\frac{63}{-8}$$

Dividing by -8 undoes the multiplication by -8.

$$x^{2}+\frac{16}{-8}x=\frac{63}{-8}$$

Divide 16 by -8.

$$x^{2}-2x=\frac{63}{-8}$$

Divide 63 by -8.

$$x^{2}-2x=-\frac{63}{8}$$

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.

$$x^{2}-2x+1=-\frac{63}{8}+1$$

Add -\frac{63}{8} to 1.

$$x^{2}-2x+1=-\frac{55}{8}$$

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}.

$$\left(x-1\right)^{2}=-\frac{55}{8}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{-\frac{55}{8}}$$

Simplify.

$$x-1=\frac{\sqrt{110}i}{4} x-1=-\frac{\sqrt{110}i}{4}$$

Add 1 to both sides of the equation.

$$x=\frac{\sqrt{110}i}{4}+1 x=-\frac{\sqrt{110}i}{4}+1$$