Solve -8x^2=-14x-15

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

Solve for x

$$x=-\frac{3}{4}=-0.75$$

$$x = \frac{5}{2} = 2\frac{1}{2} = 2.5$$

Steps Using Factoring By Grouping

Add 14x to both sides.

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

Add 15 to both sides.

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

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -8x^{2}+ax+bx+15. To find a and b, set up a system to be solved.

$$a+b=14 ab=-8\times 15=-120$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -120.

$$-1,120 -2,60 -3,40 -4,30 -5,24 -6,20 -8,15 -10,12$$

Calculate the sum for each pair.

$$-1+120=119 -2+60=58 -3+40=37 -4+30=26 -5+24=19 -6+20=14 -8+15=7 -10+12=2$$

The solution is the pair that gives sum 14.

$$a=20 b=-6$$

Rewrite -8x^{2}+14x+15 as \left(-8x^{2}+20x\right)+\left(-6x+15\right).

$$\left(-8x^{2}+20x\right)+\left(-6x+15\right)$$

Factor out -4x in the first and -3 in the second group.

$$-4x\left(2x-5\right)-3\left(2x-5\right)$$

Factor out common term 2x-5 by using distributive property.

$$\left(2x-5\right)\left(-4x-3\right)$$

To find equation solutions, solve 2x-5=0 and -4x-3=0.

$$x=\frac{5}{2} x=-\frac{3}{4}$$

Steps Using the Quadratic Formula

Add 14x to both sides.

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

Add 15 to both sides.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute -8 for a, 14 for b, and 15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}-4\left(-8\right)\times 15}}{2\left(-8\right)}$$

Square 14.

$$x=\frac{-14±\sqrt{196-4\left(-8\right)\times 15}}{2\left(-8\right)}$$

Multiply -4 times -8.

$$x=\frac{-14±\sqrt{196+32\times 15}}{2\left(-8\right)}$$

Multiply 32 times 15.

$$x=\frac{-14±\sqrt{196+480}}{2\left(-8\right)}$$

Add 196 to 480.

$$x=\frac{-14±\sqrt{676}}{2\left(-8\right)}$$

Take the square root of 676.

$$x=\frac{-14±26}{2\left(-8\right)}$$

Multiply 2 times -8.

$$x=\frac{-14±26}{-16}$$

Now solve the equation x=\frac{-14±26}{-16} when ± is plus. Add -14 to 26.

$$x=\frac{12}{-16}$$

Reduce the fraction \frac{12}{-16} to lowest terms by extracting and canceling out 4.

$$x=-\frac{3}{4}$$

Now solve the equation x=\frac{-14±26}{-16} when ± is minus. Subtract 26 from -14.

$$x=-\frac{40}{-16}$$

Reduce the fraction \frac{-40}{-16} to lowest terms by extracting and canceling out 8.

$$x=\frac{5}{2}$$

The equation is now solved.

$$x=-\frac{3}{4} x=\frac{5}{2}$$

Steps for Completing the Square

Add 14x to both sides.

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

Divide both sides by -8.

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

Dividing by -8 undoes the multiplication by -8.

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

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

$$x^{2}-\frac{7}{4}x=-\frac{15}{-8}$$

Divide -15 by -8.

$$x^{2}-\frac{7}{4}x=\frac{15}{8}$$

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.

$$x^{2}-\frac{7}{4}x+\left(-\frac{7}{8}\right)^{2}=\frac{15}{8}+\left(-\frac{7}{8}\right)^{2}$$

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

$$x^{2}-\frac{7}{4}x+\frac{49}{64}=\frac{15}{8}+\frac{49}{64}$$

Add \frac{15}{8} to \frac{49}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{7}{4}x+\frac{49}{64}=\frac{169}{64}$$

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}.

$$\left(x-\frac{7}{8}\right)^{2}=\frac{169}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{8}\right)^{2}}=\sqrt{\frac{169}{64}}$$

Simplify.

$$x-\frac{7}{8}=\frac{13}{8} x-\frac{7}{8}=-\frac{13}{8}$$

Add \frac{7}{8} to both sides of the equation.

$$x=\frac{5}{2} x=-\frac{3}{4}$$