Solve 24x^2-55x-24

$$24 x ^ { 2 } - 55 x - 24$$

Factor

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

Solution Steps

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

$$a+b=-55 ab=24\left(-24\right)=-576$$

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 -576.

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

Calculate the sum for each pair.

$$1-576=-575 2-288=-286 3-192=-189 4-144=-140 6-96=-90 8-72=-64 9-64=-55 12-48=-36 16-36=-20 18-32=-14 24-24=0$$

The solution is the pair that gives sum -55.

$$a=-64 b=9$$

Rewrite 24x^{2}-55x-24 as \left(24x^{2}-64x\right)+\left(9x-24\right).

$$\left(24x^{2}-64x\right)+\left(9x-24\right)$$

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

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

Factor out common term 3x-8 by using distributive property.

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

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.

$$24x^{2}-55x-24=0$$

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.

$$x=\frac{-\left(-55\right)±\sqrt{\left(-55\right)^{2}-4\times 24\left(-24\right)}}{2\times 24}$$

Square -55.

$$x=\frac{-\left(-55\right)±\sqrt{3025-4\times 24\left(-24\right)}}{2\times 24}$$

Multiply -4 times 24.

$$x=\frac{-\left(-55\right)±\sqrt{3025-96\left(-24\right)}}{2\times 24}$$

Multiply -96 times -24.

$$x=\frac{-\left(-55\right)±\sqrt{3025+2304}}{2\times 24}$$

Add 3025 to 2304.

$$x=\frac{-\left(-55\right)±\sqrt{5329}}{2\times 24}$$

Take the square root of 5329.

$$x=\frac{-\left(-55\right)±73}{2\times 24}$$

The opposite of -55 is 55.

$$x=\frac{55±73}{2\times 24}$$

Multiply 2 times 24.

$$x=\frac{55±73}{48}$$

Now solve the equation x=\frac{55±73}{48} when ± is plus. Add 55 to 73.

$$x=\frac{128}{48}$$

Reduce the fraction \frac{128}{48} to lowest terms by extracting and canceling out 16.

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

Now solve the equation x=\frac{55±73}{48} when ± is minus. Subtract 73 from 55.

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

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{8}{3} for x_{1} and -\frac{3}{8} for x_{2}.

$$24x^{2}-55x-24=24\left(x-\frac{8}{3}\right)\left(x-\left(-\frac{3}{8}\right)\right)$$

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

$$24x^{2}-55x-24=24\left(x-\frac{8}{3}\right)\left(x+\frac{3}{8}\right)$$

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

$$24x^{2}-55x-24=24\times \frac{3x-8}{3}\left(x+\frac{3}{8}\right)$$

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

$$24x^{2}-55x-24=24\times \frac{3x-8}{3}\times \frac{8x+3}{8}$$

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

$$24x^{2}-55x-24=24\times \frac{\left(3x-8\right)\left(8x+3\right)}{3\times 8}$$

Multiply 3 times 8.

$$24x^{2}-55x-24=24\times \frac{\left(3x-8\right)\left(8x+3\right)}{24}$$

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

$$24x^{2}-55x-24=\left(3x-8\right)\left(8x+3\right)$$

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

$$x ^ 2 -\frac{55}{24}x -1 = 0$$

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

$$r + s = \frac{55}{24} rs = -1$$

Two numbers r and s sum up to \frac{55}{24} exactly when the average of the two numbers is \frac{1}{2}*\frac{55}{24} = \frac{55}{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>

$$r = \frac{55}{48} - u s = \frac{55}{48} + u$$

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

$$(\frac{55}{48} - u) (\frac{55}{48} + u) = -1$$

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

$$\frac{3025}{2304} - u^2 = -1$$

Simplify the expression by subtracting \frac{3025}{2304} on both sides

$$-u^2 = -1-\frac{3025}{2304} = -\frac{5329}{2304}$$

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

$$u^2 = \frac{5329}{2304} u = \pm\sqrt{\frac{5329}{2304}} = \pm \frac{73}{48}$$

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

$$r =\frac{55}{48} - \frac{73}{48} = -0.375 s = \frac{55}{48} + \frac{73}{48} = 2.667$$

Evaluate

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