Solve 10x+15=8x^2+24x+78

$$10x+15 = 8 { x }^{ 2 } +24x+78$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{455}i-7}{8}\approx -0.875-2.666341126i$$

$$x=\frac{-7+\sqrt{455}i}{8}\approx -0.875+2.666341126i$$

Steps Using the Quadratic Formula

Subtract 8x^{2} from both sides.

$$10x+15-8x^{2}=24x+78$$

Subtract 24x from both sides.

$$10x+15-8x^{2}-24x=78$$

Combine 10x and -24x to get -14x.

$$-14x+15-8x^{2}=78$$

Subtract 78 from both sides.

$$-14x+15-8x^{2}-78=0$$

Subtract 78 from 15 to get -63.

$$-14x-63-8x^{2}=0$$

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.

$$-8x^{2}-14x-63=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -8 for a, -14 for b, and -63 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\left(-8\right)\left(-63\right)}}{2\left(-8\right)}$$

Square -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\left(-8\right)\left(-63\right)}}{2\left(-8\right)}$$

Multiply -4 times -8.

$$x=\frac{-\left(-14\right)±\sqrt{196+32\left(-63\right)}}{2\left(-8\right)}$$

Multiply 32 times -63.

$$x=\frac{-\left(-14\right)±\sqrt{196-2016}}{2\left(-8\right)}$$

Add 196 to -2016.

$$x=\frac{-\left(-14\right)±\sqrt{-1820}}{2\left(-8\right)}$$

Take the square root of -1820.

$$x=\frac{-\left(-14\right)±2\sqrt{455}i}{2\left(-8\right)}$$

The opposite of -14 is 14.

$$x=\frac{14±2\sqrt{455}i}{2\left(-8\right)}$$

Multiply 2 times -8.

$$x=\frac{14±2\sqrt{455}i}{-16}$$

Now solve the equation x=\frac{14±2\sqrt{455}i}{-16} when ± is plus. Add 14 to 2i\sqrt{455}.

$$x=\frac{14+2\sqrt{455}i}{-16}$$

Divide 14+2i\sqrt{455} by -16.

$$x=\frac{-\sqrt{455}i-7}{8}$$

Now solve the equation x=\frac{14±2\sqrt{455}i}{-16} when ± is minus. Subtract 2i\sqrt{455} from 14.

$$x=\frac{-2\sqrt{455}i+14}{-16}$$

Divide 14-2i\sqrt{455} by -16.

$$x=\frac{-7+\sqrt{455}i}{8}$$

The equation is now solved.

$$x=\frac{-\sqrt{455}i-7}{8} x=\frac{-7+\sqrt{455}i}{8}$$

Steps for Completing the Square

Subtract 8x^{2} from both sides.

$$10x+15-8x^{2}=24x+78$$

Subtract 24x from both sides.

$$10x+15-8x^{2}-24x=78$$

Combine 10x and -24x to get -14x.

$$-14x+15-8x^{2}=78$$

Subtract 15 from both sides.

$$-14x-8x^{2}=78-15$$

Subtract 15 from 78 to get 63.

$$-14x-8x^{2}=63$$

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.

$$-8x^{2}-14x=63$$

Divide both sides by -8.

$$\frac{-8x^{2}-14x}{-8}=\frac{63}{-8}$$

Dividing by -8 undoes the multiplication by -8.

$$x^{2}+\left(-\frac{14}{-8}\right)x=\frac{63}{-8}$$

Reduce the fraction \frac{-14}{-8} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{7}{4}x=\frac{63}{-8}$$

Divide 63 by -8.

$$x^{2}+\frac{7}{4}x=-\frac{63}{8}$$

Divide \frac{7}{4}, the coefficient of the x term, by 2 to get \frac{7}{8}. Then add the square of \frac{7}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{7}{4}x+\left(\frac{7}{8}\right)^{2}=-\frac{63}{8}+\left(\frac{7}{8}\right)^{2}$$

Square \frac{7}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{7}{4}x+\frac{49}{64}=-\frac{63}{8}+\frac{49}{64}$$

Add -\frac{63}{8} to \frac{49}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{7}{4}x+\frac{49}{64}=-\frac{455}{64}$$

Factor x^{2}+\frac{7}{4}x+\frac{49}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{7}{8}\right)^{2}=-\frac{455}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{8}\right)^{2}}=\sqrt{-\frac{455}{64}}$$

Simplify.

$$x+\frac{7}{8}=\frac{\sqrt{455}i}{8} x+\frac{7}{8}=-\frac{\sqrt{455}i}{8}$$

Subtract \frac{7}{8} from both sides of the equation.

$$x=\frac{-7+\sqrt{455}i}{8} x=\frac{-\sqrt{455}i-7}{8}$$