Solve 18x^2-33x-216

$$18 x ^ { 2 } - 33 x - 216$$

Factor

$$3\left(2x-9\right)\left(3x+8\right)$$

Solution Steps

Factor out 3.

$$3\left(6x^{2}-11x-72\right)$$

Consider 6x^{2}-11x-72. Factor the expression by grouping. First, the expression needs to be rewritten as 6x^{2}+ax+bx-72. To find a and b, set up a system to be solved.

$$a+b=-11 ab=6\left(-72\right)=-432$$

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 -432.

$$1,-432 2,-216 3,-144 4,-108 6,-72 8,-54 9,-48 12,-36 16,-27 18,-24$$

Calculate the sum for each pair.

$$1-432=-431 2-216=-214 3-144=-141 4-108=-104 6-72=-66 8-54=-46 9-48=-39 12-36=-24 16-27=-11 18-24=-6$$

The solution is the pair that gives sum -11.

$$a=-27 b=16$$

Rewrite 6x^{2}-11x-72 as \left(6x^{2}-27x\right)+\left(16x-72\right).

$$\left(6x^{2}-27x\right)+\left(16x-72\right)$$

Factor out 3x in the first and 8 in the second group.

$$3x\left(2x-9\right)+8\left(2x-9\right)$$

Factor out common term 2x-9 by using distributive property.

$$\left(2x-9\right)\left(3x+8\right)$$

Rewrite the complete factored expression.

$$3\left(2x-9\right)\left(3x+8\right)$$

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.

$$18x^{2}-33x-216=0$$

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.

$$x=\frac{-\left(-33\right)±\sqrt{\left(-33\right)^{2}-4\times 18\left(-216\right)}}{2\times 18}$$

Square -33.

$$x=\frac{-\left(-33\right)±\sqrt{1089-4\times 18\left(-216\right)}}{2\times 18}$$

Multiply -4 times 18.

$$x=\frac{-\left(-33\right)±\sqrt{1089-72\left(-216\right)}}{2\times 18}$$

Multiply -72 times -216.

$$x=\frac{-\left(-33\right)±\sqrt{1089+15552}}{2\times 18}$$

Add 1089 to 15552.

$$x=\frac{-\left(-33\right)±\sqrt{16641}}{2\times 18}$$

Take the square root of 16641.

$$x=\frac{-\left(-33\right)±129}{2\times 18}$$

The opposite of -33 is 33.

$$x=\frac{33±129}{2\times 18}$$

Multiply 2 times 18.

$$x=\frac{33±129}{36}$$

Now solve the equation x=\frac{33±129}{36} when ± is plus. Add 33 to 129.

$$x=\frac{162}{36}$$

Reduce the fraction \frac{162}{36} to lowest terms by extracting and canceling out 18.

$$x=\frac{9}{2}$$

Now solve the equation x=\frac{33±129}{36} when ± is minus. Subtract 129 from 33.

$$x=-\frac{96}{36}$$

Reduce the fraction \frac{-96}{36} to lowest terms by extracting and canceling out 12.

$$x=-\frac{8}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{9}{2} for x_{1} and -\frac{8}{3} for x_{2}.

$$18x^{2}-33x-216=18\left(x-\frac{9}{2}\right)\left(x-\left(-\frac{8}{3}\right)\right)$$

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

$$18x^{2}-33x-216=18\left(x-\frac{9}{2}\right)\left(x+\frac{8}{3}\right)$$

Subtract \frac{9}{2} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$18x^{2}-33x-216=18\times \frac{2x-9}{2}\left(x+\frac{8}{3}\right)$$

Add \frac{8}{3} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$18x^{2}-33x-216=18\times \frac{2x-9}{2}\times \frac{3x+8}{3}$$

Multiply \frac{2x-9}{2} times \frac{3x+8}{3} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$18x^{2}-33x-216=18\times \frac{\left(2x-9\right)\left(3x+8\right)}{2\times 3}$$

Multiply 2 times 3.

$$18x^{2}-33x-216=18\times \frac{\left(2x-9\right)\left(3x+8\right)}{6}$$

Cancel out 6, the greatest common factor in 18 and 6.

$$18x^{2}-33x-216=3\left(2x-9\right)\left(3x+8\right)$$

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 18

$$x ^ 2 -\frac{11}{6}x -12 = 0$$

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

$$r + s = \frac{11}{6} rs = -12$$

Two numbers r and s sum up to \frac{11}{6} exactly when the average of the two numbers is \frac{1}{2}*\frac{11}{6} = \frac{11}{12}. 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>

$$r = \frac{11}{12} - u s = \frac{11}{12} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -12

$$(\frac{11}{12} - u) (\frac{11}{12} + u) = -12$$

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

$$\frac{121}{144} - u^2 = -12$$

Simplify the expression by subtracting \frac{121}{144} on both sides

$$-u^2 = -12-\frac{121}{144} = -\frac{1849}{144}$$

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

$$u^2 = \frac{1849}{144} u = \pm\sqrt{\frac{1849}{144}} = \pm \frac{43}{12}$$

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

$$r =\frac{11}{12} - \frac{43}{12} = -2.667 s = \frac{11}{12} + \frac{43}{12} = 4.500$$

Evaluate

$$18x^{2}-33x-216$$