Solve 4x^2-10x-24

$$4 { x }^{ 2 } -10x-24$$

Factor

$$2\left(x-4\right)\left(2x+3\right)$$

Solution Steps

Factor out 2.

$$2\left(2x^{2}-5x-12\right)$$

Consider 2x^{2}-5x-12. Factor the expression by grouping. First, the expression needs to be rewritten as 2x^{2}+ax+bx-12. To find a and b, set up a system to be solved.

$$a+b=-5 ab=2\left(-12\right)=-24$$

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 -24.

$$1,-24 2,-12 3,-8 4,-6$$

Calculate the sum for each pair.

$$1-24=-23 2-12=-10 3-8=-5 4-6=-2$$

The solution is the pair that gives sum -5.

$$a=-8 b=3$$

Rewrite 2x^{2}-5x-12 as \left(2x^{2}-8x\right)+\left(3x-12\right).

$$\left(2x^{2}-8x\right)+\left(3x-12\right)$$

Factor out 2x in the first and 3 in the second group.

$$2x\left(x-4\right)+3\left(x-4\right)$$

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

$$\left(x-4\right)\left(2x+3\right)$$

Rewrite the complete factored expression.

$$2\left(x-4\right)\left(2x+3\right)$$

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.

$$4x^{2}-10x-24=0$$

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.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 4\left(-24\right)}}{2\times 4}$$

Square -10.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\times 4\left(-24\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-10\right)±\sqrt{100-16\left(-24\right)}}{2\times 4}$$

Multiply -16 times -24.

$$x=\frac{-\left(-10\right)±\sqrt{100+384}}{2\times 4}$$

Add 100 to 384.

$$x=\frac{-\left(-10\right)±\sqrt{484}}{2\times 4}$$

Take the square root of 484.

$$x=\frac{-\left(-10\right)±22}{2\times 4}$$

The opposite of -10 is 10.

$$x=\frac{10±22}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{10±22}{8}$$

Now solve the equation x=\frac{10±22}{8} when ± is plus. Add 10 to 22.

$$x=\frac{32}{8}$$

Divide 32 by 8.

$$x=4$$

Now solve the equation x=\frac{10±22}{8} when ± is minus. Subtract 22 from 10.

$$x=-\frac{12}{8}$$

Reduce the fraction \frac{-12}{8} to lowest terms by extracting and canceling out 4.

$$x=-\frac{3}{2}$$

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 -\frac{3}{2} for x_{2}.

$$4x^{2}-10x-24=4\left(x-4\right)\left(x-\left(-\frac{3}{2}\right)\right)$$

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

$$4x^{2}-10x-24=4\left(x-4\right)\left(x+\frac{3}{2}\right)$$

Add \frac{3}{2} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$4x^{2}-10x-24=4\left(x-4\right)\times \frac{2x+3}{2}$$

Cancel out 2, the greatest common factor in 4 and 2.

$$4x^{2}-10x-24=2\left(x-4\right)\left(2x+3\right)$$

Evaluate

$$2\left(x-4\right)\left(2x+3\right)$$