Solve 8m^2-14m-15

$$8 m ^ { 2 } - 14 m - 15$$

Factor

$$\left(2m-5\right)\left(4m+3\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 8m^{2}+am+bm-15. To find a and b, set up a system to be solved.

$$a+b=-14 ab=8\left(-15\right)=-120$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. 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 8m^{2}-14m-15 as \left(8m^{2}-20m\right)+\left(6m-15\right).

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

Factor out 4m in the first and 3 in the second group.

$$4m\left(2m-5\right)+3\left(2m-5\right)$$

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

$$\left(2m-5\right)\left(4m+3\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$8m^{2}-14m-15=0$$

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.

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

Square -14.

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

Multiply -4 times 8.

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

Multiply -32 times -15.

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

Add 196 to 480.

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

Take the square root of 676.

$$m=\frac{-\left(-14\right)±26}{2\times 8}$$

The opposite of -14 is 14.

$$m=\frac{14±26}{2\times 8}$$

Multiply 2 times 8.

$$m=\frac{14±26}{16}$$

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

$$m=\frac{40}{16}$$

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

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

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

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

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

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

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{5}{2} for x_{1} and -\frac{3}{4} for x_{2}.

$$8m^{2}-14m-15=8\left(m-\frac{5}{2}\right)\left(m-\left(-\frac{3}{4}\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$8m^{2}-14m-15=8\left(m-\frac{5}{2}\right)\left(m+\frac{3}{4}\right)$$

Subtract \frac{5}{2} from m by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$8m^{2}-14m-15=8\times \frac{2m-5}{2}\left(m+\frac{3}{4}\right)$$

Add \frac{3}{4} to m by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$8m^{2}-14m-15=8\times \frac{2m-5}{2}\times \frac{4m+3}{4}$$

Multiply \frac{2m-5}{2} times \frac{4m+3}{4} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$8m^{2}-14m-15=8\times \frac{\left(2m-5\right)\left(4m+3\right)}{2\times 4}$$

Multiply 2 times 4.

$$8m^{2}-14m-15=8\times \frac{\left(2m-5\right)\left(4m+3\right)}{8}$$

Cancel out 8, the greatest common factor in 8 and 8.

$$8m^{2}-14m-15=\left(2m-5\right)\left(4m+3\right)$$

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.This is achieved by dividing both sides of the equation by 8

$$x ^ 2 -\frac{7}{4}x -\frac{15}{8} = 0$$

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = \frac{7}{4} rs = -\frac{15}{8}$$

Two numbers r and s sum up to \frac{7}{4} exactly when the average of the two numbers is \frac{1}{2}*\frac{7}{4} = \frac{7}{8}. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = \frac{7}{8} - u s = \frac{7}{8} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{15}{8}

$$(\frac{7}{8} - u) (\frac{7}{8} + u) = -\frac{15}{8}$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$\frac{49}{64} - u^2 = -\frac{15}{8}$$

Simplify the expression by subtracting \frac{49}{64} on both sides

$$-u^2 = -\frac{15}{8}-\frac{49}{64} = -\frac{169}{64}$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = \frac{169}{64} u = \pm\sqrt{\frac{169}{64}} = \pm \frac{13}{8}$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =\frac{7}{8} - \frac{13}{8} = -0.750 s = \frac{7}{8} + \frac{13}{8} = 2.500$$

Evaluate

$$\left(2m-5\right)\left(4m+3\right)$$