Solve 24x^2-119x-48

$$24 x ^ { 2 } - 119 x - 48$$

Factor

$$\left(3x-16\right)\left(8x+3\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 24x^{2}+ax+bx-48. To find a and b, set up a system to be solved.

$$a+b=-119 ab=24\left(-48\right)=-1152$$

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 -1152.

$$1,-1152 2,-576 3,-384 4,-288 6,-192 8,-144 9,-128 12,-96 16,-72 18,-64 24,-48 32,-36$$

Calculate the sum for each pair.

$$1-1152=-1151 2-576=-574 3-384=-381 4-288=-284 6-192=-186 8-144=-136 9-128=-119 12-96=-84 16-72=-56 18-64=-46 24-48=-24 32-36=-4$$

The solution is the pair that gives sum -119.

$$a=-128 b=9$$

Rewrite 24x^{2}-119x-48 as \left(24x^{2}-128x\right)+\left(9x-48\right).

$$\left(24x^{2}-128x\right)+\left(9x-48\right)$$

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

$$8x\left(3x-16\right)+3\left(3x-16\right)$$

Factor out common term 3x-16 by using distributive property.

$$\left(3x-16\right)\left(8x+3\right)$$

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.

$$24x^{2}-119x-48=0$$

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.

$$x=\frac{-\left(-119\right)±\sqrt{\left(-119\right)^{2}-4\times 24\left(-48\right)}}{2\times 24}$$

Square -119.

$$x=\frac{-\left(-119\right)±\sqrt{14161-4\times 24\left(-48\right)}}{2\times 24}$$

Multiply -4 times 24.

$$x=\frac{-\left(-119\right)±\sqrt{14161-96\left(-48\right)}}{2\times 24}$$

Multiply -96 times -48.

$$x=\frac{-\left(-119\right)±\sqrt{14161+4608}}{2\times 24}$$

Add 14161 to 4608.

$$x=\frac{-\left(-119\right)±\sqrt{18769}}{2\times 24}$$

Take the square root of 18769.

$$x=\frac{-\left(-119\right)±137}{2\times 24}$$

The opposite of -119 is 119.

$$x=\frac{119±137}{2\times 24}$$

Multiply 2 times 24.

$$x=\frac{119±137}{48}$$

Now solve the equation x=\frac{119±137}{48} when ± is plus. Add 119 to 137.

$$x=\frac{256}{48}$$

Reduce the fraction \frac{256}{48} to lowest terms by extracting and canceling out 16.

$$x=\frac{16}{3}$$

Now solve the equation x=\frac{119±137}{48} when ± is minus. Subtract 137 from 119.

$$x=-\frac{18}{48}$$

Reduce the fraction \frac{-18}{48} to lowest terms by extracting and canceling out 6.

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

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{16}{3} for x_{1} and -\frac{3}{8} for x_{2}.

$$24x^{2}-119x-48=24\left(x-\frac{16}{3}\right)\left(x-\left(-\frac{3}{8}\right)\right)$$

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

$$24x^{2}-119x-48=24\left(x-\frac{16}{3}\right)\left(x+\frac{3}{8}\right)$$

Subtract \frac{16}{3} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$24x^{2}-119x-48=24\times \frac{3x-16}{3}\left(x+\frac{3}{8}\right)$$

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

$$24x^{2}-119x-48=24\times \frac{3x-16}{3}\times \frac{8x+3}{8}$$

Multiply \frac{3x-16}{3} times \frac{8x+3}{8} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$24x^{2}-119x-48=24\times \frac{\left(3x-16\right)\left(8x+3\right)}{3\times 8}$$

Multiply 3 times 8.

$$24x^{2}-119x-48=24\times \frac{\left(3x-16\right)\left(8x+3\right)}{24}$$

Cancel out 24, the greatest common factor in 24 and 24.

$$24x^{2}-119x-48=\left(3x-16\right)\left(8x+3\right)$$

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 24

$$x ^ 2 -\frac{119}{24}x -2 = 0$$

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

$$r + s = \frac{119}{24} rs = -2$$

Two numbers r and s sum up to \frac{119}{24} exactly when the average of the two numbers is \frac{1}{2}*\frac{119}{24} = \frac{119}{48}. 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>

$$r = \frac{119}{48} - u s = \frac{119}{48} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -2

$$(\frac{119}{48} - u) (\frac{119}{48} + u) = -2$$

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

$$\frac{14161}{2304} - u^2 = -2$$

Simplify the expression by subtracting \frac{14161}{2304} on both sides

$$-u^2 = -2-\frac{14161}{2304} = -\frac{18769}{2304}$$

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

$$u^2 = \frac{18769}{2304} u = \pm\sqrt{\frac{18769}{2304}} = \pm \frac{137}{48}$$

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

$$r =\frac{119}{48} - \frac{137}{48} = -0.375 s = \frac{119}{48} + \frac{137}{48} = 5.333$$

Evaluate

$$\left(3x-16\right)\left(8x+3\right)$$