Solve 12x^2-7x-12

$$12 x ^ { 2 } - 7 x - 12$$

Factor

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

Solution Steps

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

$$a+b=-7 ab=12\left(-12\right)=-144$$

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 -144.

$$1,-144 2,-72 3,-48 4,-36 6,-24 8,-18 9,-16 12,-12$$

Calculate the sum for each pair.

$$1-144=-143 2-72=-70 3-48=-45 4-36=-32 6-24=-18 8-18=-10 9-16=-7 12-12=0$$

The solution is the pair that gives sum -7.

$$a=-16 b=9$$

Rewrite 12x^{2}-7x-12 as \left(12x^{2}-16x\right)+\left(9x-12\right).

$$\left(12x^{2}-16x\right)+\left(9x-12\right)$$

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

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

Factor out common term 3x-4 by using distributive property.

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

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.

$$12x^{2}-7x-12=0$$

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.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 12\left(-12\right)}}{2\times 12}$$

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 12\left(-12\right)}}{2\times 12}$$

Multiply -4 times 12.

$$x=\frac{-\left(-7\right)±\sqrt{49-48\left(-12\right)}}{2\times 12}$$

Multiply -48 times -12.

$$x=\frac{-\left(-7\right)±\sqrt{49+576}}{2\times 12}$$

Add 49 to 576.

$$x=\frac{-\left(-7\right)±\sqrt{625}}{2\times 12}$$

Take the square root of 625.

$$x=\frac{-\left(-7\right)±25}{2\times 12}$$

The opposite of -7 is 7.

$$x=\frac{7±25}{2\times 12}$$

Multiply 2 times 12.

$$x=\frac{7±25}{24}$$

Now solve the equation x=\frac{7±25}{24} when ± is plus. Add 7 to 25.

$$x=\frac{32}{24}$$

Reduce the fraction \frac{32}{24} to lowest terms by extracting and canceling out 8.

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

Now solve the equation x=\frac{7±25}{24} when ± is minus. Subtract 25 from 7.

$$x=-\frac{18}{24}$$

Reduce the fraction \frac{-18}{24} to lowest terms by extracting and canceling out 6.

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

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

$$12x^{2}-7x-12=12\left(x-\frac{4}{3}\right)\left(x-\left(-\frac{3}{4}\right)\right)$$

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

$$12x^{2}-7x-12=12\left(x-\frac{4}{3}\right)\left(x+\frac{3}{4}\right)$$

Subtract \frac{4}{3} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$12x^{2}-7x-12=12\times \frac{3x-4}{3}\left(x+\frac{3}{4}\right)$$

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

$$12x^{2}-7x-12=12\times \frac{3x-4}{3}\times \frac{4x+3}{4}$$

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

$$12x^{2}-7x-12=12\times \frac{\left(3x-4\right)\left(4x+3\right)}{3\times 4}$$

Multiply 3 times 4.

$$12x^{2}-7x-12=12\times \frac{\left(3x-4\right)\left(4x+3\right)}{12}$$

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

$$12x^{2}-7x-12=\left(3x-4\right)\left(4x+3\right)$$

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 12

$$x ^ 2 -\frac{7}{12}x -1 = 0$$

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

$$r + s = \frac{7}{12} rs = -1$$

Two numbers r and s sum up to \frac{7}{12} exactly when the average of the two numbers is \frac{1}{2}*\frac{7}{12} = \frac{7}{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>

$$r = \frac{7}{24} - u s = \frac{7}{24} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -1

$$(\frac{7}{24} - u) (\frac{7}{24} + u) = -1$$

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

$$\frac{49}{576} - u^2 = -1$$

Simplify the expression by subtracting \frac{49}{576} on both sides

$$-u^2 = -1-\frac{49}{576} = -\frac{625}{576}$$

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

$$u^2 = \frac{625}{576} u = \pm\sqrt{\frac{625}{576}} = \pm \frac{25}{24}$$

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

$$r =\frac{7}{24} - \frac{25}{24} = -0.750 s = \frac{7}{24} + \frac{25}{24} = 1.333$$

Evaluate

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