Solve 12h^2-31h-15

$$12 h ^ { 2 } - 31 h - 15$$

Factor

$$\left(h-3\right)\left(12h+5\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 12h^{2}+ah+bh-15. To find a and b, set up a system to be solved.

$$a+b=-31 ab=12\left(-15\right)=-180$$

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 -180.

$$1,-180 2,-90 3,-60 4,-45 5,-36 6,-30 9,-20 10,-18 12,-15$$

Calculate the sum for each pair.

$$1-180=-179 2-90=-88 3-60=-57 4-45=-41 5-36=-31 6-30=-24 9-20=-11 10-18=-8 12-15=-3$$

The solution is the pair that gives sum -31.

$$a=-36 b=5$$

Rewrite 12h^{2}-31h-15 as \left(12h^{2}-36h\right)+\left(5h-15\right).

$$\left(12h^{2}-36h\right)+\left(5h-15\right)$$

Factor out 12h in the first and 5 in the second group.

$$12h\left(h-3\right)+5\left(h-3\right)$$

Factor out common term h-3 by using distributive property.

$$\left(h-3\right)\left(12h+5\right)$$

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.

$$12h^{2}-31h-15=0$$

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.

$$h=\frac{-\left(-31\right)±\sqrt{\left(-31\right)^{2}-4\times 12\left(-15\right)}}{2\times 12}$$

Square -31.

$$h=\frac{-\left(-31\right)±\sqrt{961-4\times 12\left(-15\right)}}{2\times 12}$$

Multiply -4 times 12.

$$h=\frac{-\left(-31\right)±\sqrt{961-48\left(-15\right)}}{2\times 12}$$

Multiply -48 times -15.

$$h=\frac{-\left(-31\right)±\sqrt{961+720}}{2\times 12}$$

Add 961 to 720.

$$h=\frac{-\left(-31\right)±\sqrt{1681}}{2\times 12}$$

Take the square root of 1681.

$$h=\frac{-\left(-31\right)±41}{2\times 12}$$

The opposite of -31 is 31.

$$h=\frac{31±41}{2\times 12}$$

Multiply 2 times 12.

$$h=\frac{31±41}{24}$$

Now solve the equation h=\frac{31±41}{24} when ± is plus. Add 31 to 41.

$$h=\frac{72}{24}$$

Divide 72 by 24.

$$h=3$$

Now solve the equation h=\frac{31±41}{24} when ± is minus. Subtract 41 from 31.

$$h=-\frac{10}{24}$$

Reduce the fraction \frac{-10}{24} to lowest terms by extracting and canceling out 2.

$$h=-\frac{5}{12}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 3 for x_{1} and -\frac{5}{12} for x_{2}.

$$12h^{2}-31h-15=12\left(h-3\right)\left(h-\left(-\frac{5}{12}\right)\right)$$

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

$$12h^{2}-31h-15=12\left(h-3\right)\left(h+\frac{5}{12}\right)$$

Add \frac{5}{12} to h by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$12h^{2}-31h-15=12\left(h-3\right)\times \frac{12h+5}{12}$$

Cancel out 12, the greatest common factor in 12 and 12.

$$12h^{2}-31h-15=\left(h-3\right)\left(12h+5\right)$$

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 12

$$x ^ 2 -\frac{31}{12}x -\frac{5}{4} = 0$$

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

$$r + s = \frac{31}{12} rs = -\frac{5}{4}$$

Two numbers r and s sum up to \frac{31}{12} exactly when the average of the two numbers is \frac{1}{2}*\frac{31}{12} = \frac{31}{24}. 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>

$$r = \frac{31}{24} - u s = \frac{31}{24} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{5}{4}

$$(\frac{31}{24} - u) (\frac{31}{24} + u) = -\frac{5}{4}$$

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

$$\frac{961}{576} - u^2 = -\frac{5}{4}$$

Simplify the expression by subtracting \frac{961}{576} on both sides

$$-u^2 = -\frac{5}{4}-\frac{961}{576} = -\frac{1681}{576}$$

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

$$u^2 = \frac{1681}{576} u = \pm\sqrt{\frac{1681}{576}} = \pm \frac{41}{24}$$

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

$$r =\frac{31}{24} - \frac{41}{24} = -0.417 s = \frac{31}{24} + \frac{41}{24} = 3$$

Evaluate

$$\left(h-3\right)\left(12h+5\right)$$