Solve 7x^2-53x-24

$$7 { x }^{ 2 } -53x-24$$

Factor

$$\left(x-8\right)\left(7x+3\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 7x^{2}+ax+bx-24. To find a and b, set up a system to be solved.

$$a+b=-53 ab=7\left(-24\right)=-168$$

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 -168.

$$1,-168 2,-84 3,-56 4,-42 6,-28 7,-24 8,-21 12,-14$$

Calculate the sum for each pair.

$$1-168=-167 2-84=-82 3-56=-53 4-42=-38 6-28=-22 7-24=-17 8-21=-13 12-14=-2$$

The solution is the pair that gives sum -53.

$$a=-56 b=3$$

Rewrite 7x^{2}-53x-24 as \left(7x^{2}-56x\right)+\left(3x-24\right).

$$\left(7x^{2}-56x\right)+\left(3x-24\right)$$

Factor out 7x in the first and 3 in the second group.

$$7x\left(x-8\right)+3\left(x-8\right)$$

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

$$\left(x-8\right)\left(7x+3\right)$$

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.

$$7x^{2}-53x-24=0$$

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.

$$x=\frac{-\left(-53\right)±\sqrt{\left(-53\right)^{2}-4\times 7\left(-24\right)}}{2\times 7}$$

Square -53.

$$x=\frac{-\left(-53\right)±\sqrt{2809-4\times 7\left(-24\right)}}{2\times 7}$$

Multiply -4 times 7.

$$x=\frac{-\left(-53\right)±\sqrt{2809-28\left(-24\right)}}{2\times 7}$$

Multiply -28 times -24.

$$x=\frac{-\left(-53\right)±\sqrt{2809+672}}{2\times 7}$$

Add 2809 to 672.

$$x=\frac{-\left(-53\right)±\sqrt{3481}}{2\times 7}$$

Take the square root of 3481.

$$x=\frac{-\left(-53\right)±59}{2\times 7}$$

The opposite of -53 is 53.

$$x=\frac{53±59}{2\times 7}$$

Multiply 2 times 7.

$$x=\frac{53±59}{14}$$

Now solve the equation x=\frac{53±59}{14} when ± is plus. Add 53 to 59.

$$x=\frac{112}{14}$$

Divide 112 by 14.

$$x=8$$

Now solve the equation x=\frac{53±59}{14} when ± is minus. Subtract 59 from 53.

$$x=-\frac{6}{14}$$

Reduce the fraction \frac{-6}{14} to lowest terms by extracting and canceling out 2.

$$x=-\frac{3}{7}$$

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}{7} for x_{2}.

$$7x^{2}-53x-24=7\left(x-8\right)\left(x-\left(-\frac{3}{7}\right)\right)$$

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

$$7x^{2}-53x-24=7\left(x-8\right)\left(x+\frac{3}{7}\right)$$

Add \frac{3}{7} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$7x^{2}-53x-24=7\left(x-8\right)\times \frac{7x+3}{7}$$

Cancel out 7, the greatest common factor in 7 and 7.

$$7x^{2}-53x-24=\left(x-8\right)\left(7x+3\right)$$

Evaluate

$$\left(x-8\right)\left(7x+3\right)$$