Solve 9x^2-9x-108

$$9 x ^ { 2 } - 9 x - 108$$

Factor

$$9\left(x-4\right)\left(x+3\right)$$

Solution Steps

Factor out 9.

$$9\left(x^{2}-x-12\right)$$

Consider x^{2}-x-12. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-12. To find a and b, set up a system to be solved.

$$a+b=-1 ab=1\left(-12\right)=-12$$

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 -12.

$$1,-12 2,-6 3,-4$$

Calculate the sum for each pair.

$$1-12=-11 2-6=-4 3-4=-1$$

The solution is the pair that gives sum -1.

$$a=-4 b=3$$

Rewrite x^{2}-x-12 as \left(x^{2}-4x\right)+\left(3x-12\right).

$$\left(x^{2}-4x\right)+\left(3x-12\right)$$

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

$$x\left(x-4\right)+3\left(x-4\right)$$

Factor out common term x-4 by using distributive property.

$$\left(x-4\right)\left(x+3\right)$$

Rewrite the complete factored expression.

$$9\left(x-4\right)\left(x+3\right)$$

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.

$$9x^{2}-9x-108=0$$

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.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 9\left(-108\right)}}{2\times 9}$$

Square -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\times 9\left(-108\right)}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-\left(-9\right)±\sqrt{81-36\left(-108\right)}}{2\times 9}$$

Multiply -36 times -108.

$$x=\frac{-\left(-9\right)±\sqrt{81+3888}}{2\times 9}$$

Add 81 to 3888.

$$x=\frac{-\left(-9\right)±\sqrt{3969}}{2\times 9}$$

Take the square root of 3969.

$$x=\frac{-\left(-9\right)±63}{2\times 9}$$

The opposite of -9 is 9.

$$x=\frac{9±63}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{9±63}{18}$$

Now solve the equation x=\frac{9±63}{18} when ± is plus. Add 9 to 63.

$$x=\frac{72}{18}$$

Divide 72 by 18.

$$x=4$$

Now solve the equation x=\frac{9±63}{18} when ± is minus. Subtract 63 from 9.

$$x=-\frac{54}{18}$$

Divide -54 by 18.

$$x=-3$$

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

$$9x^{2}-9x-108=9\left(x-4\right)\left(x-\left(-3\right)\right)$$

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

$$9x^{2}-9x-108=9\left(x-4\right)\left(x+3\right)$$

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 9

$$x ^ 2 -1x -12 = 0$$

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

$$r + s = 1 rs = -12$$

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

$$r = \frac{1}{2} - u s = \frac{1}{2} + u$$

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

$$(\frac{1}{2} - u) (\frac{1}{2} + u) = -12$$

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

$$\frac{1}{4} - u^2 = -12$$

Simplify the expression by subtracting \frac{1}{4} on both sides

$$-u^2 = -12-\frac{1}{4} = -\frac{49}{4}$$

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

$$u^2 = \frac{49}{4} u = \pm\sqrt{\frac{49}{4}} = \pm \frac{7}{2}$$

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

$$r =\frac{1}{2} - \frac{7}{2} = -3 s = \frac{1}{2} + \frac{7}{2} = 4$$

Evaluate

$$9\left(x-4\right)\left(x+3\right)$$