Solve 4x^2-29x-24

$$4 { x }^{ 2 } -29x-24$$

Factor

$$\left(x-8\right)\left(4x+3\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 4x^{2}+ax+bx-24. To find a and b, set up a system to be solved.

$$a+b=-29 ab=4\left(-24\right)=-96$$

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 -96.

$$1,-96 2,-48 3,-32 4,-24 6,-16 8,-12$$

Calculate the sum for each pair.

$$1-96=-95 2-48=-46 3-32=-29 4-24=-20 6-16=-10 8-12=-4$$

The solution is the pair that gives sum -29.

$$a=-32 b=3$$

Rewrite 4x^{2}-29x-24 as \left(4x^{2}-32x\right)+\left(3x-24\right).

$$\left(4x^{2}-32x\right)+\left(3x-24\right)$$

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

$$4x\left(x-8\right)+3\left(x-8\right)$$

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

$$\left(x-8\right)\left(4x+3\right)$$

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.

$$4x^{2}-29x-24=0$$

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.

$$x=\frac{-\left(-29\right)±\sqrt{\left(-29\right)^{2}-4\times 4\left(-24\right)}}{2\times 4}$$

Square -29.

$$x=\frac{-\left(-29\right)±\sqrt{841-4\times 4\left(-24\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-29\right)±\sqrt{841-16\left(-24\right)}}{2\times 4}$$

Multiply -16 times -24.

$$x=\frac{-\left(-29\right)±\sqrt{841+384}}{2\times 4}$$

Add 841 to 384.

$$x=\frac{-\left(-29\right)±\sqrt{1225}}{2\times 4}$$

Take the square root of 1225.

$$x=\frac{-\left(-29\right)±35}{2\times 4}$$

The opposite of -29 is 29.

$$x=\frac{29±35}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{29±35}{8}$$

Now solve the equation x=\frac{29±35}{8} when ± is plus. Add 29 to 35.

$$x=\frac{64}{8}$$

Divide 64 by 8.

$$x=8$$

Now solve the equation x=\frac{29±35}{8} when ± is minus. Subtract 35 from 29.

$$x=-\frac{6}{8}$$

Reduce the fraction \frac{-6}{8} to lowest terms by extracting and canceling out 2.

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

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 8 for x_{1} and -\frac{3}{4} for x_{2}.

$$4x^{2}-29x-24=4\left(x-8\right)\left(x-\left(-\frac{3}{4}\right)\right)$$

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

$$4x^{2}-29x-24=4\left(x-8\right)\left(x+\frac{3}{4}\right)$$

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

$$4x^{2}-29x-24=4\left(x-8\right)\times \frac{4x+3}{4}$$

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

$$4x^{2}-29x-24=\left(x-8\right)\left(4x+3\right)$$

Evaluate

$$\left(x-8\right)\left(4x+3\right)$$