Solve 3x^2-53x+232=

$$3 { x }^{ 2 } -53x+232=$$

Factor

$$\left(x-8\right)\left(3x-29\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 3x^{2}+ax+bx+232. To find a and b, set up a system to be solved.

$$a+b=-53 ab=3\times 232=696$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 696.

$$-1,-696 -2,-348 -3,-232 -4,-174 -6,-116 -8,-87 -12,-58 -24,-29$$

Calculate the sum for each pair.

$$-1-696=-697 -2-348=-350 -3-232=-235 -4-174=-178 -6-116=-122 -8-87=-95 -12-58=-70 -24-29=-53$$

The solution is the pair that gives sum -53.

$$a=-29 b=-24$$

Rewrite 3x^{2}-53x+232 as \left(3x^{2}-29x\right)+\left(-24x+232\right).

$$\left(3x^{2}-29x\right)+\left(-24x+232\right)$$

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

$$x\left(3x-29\right)-8\left(3x-29\right)$$

Factor out common term 3x-29 by using distributive property.

$$\left(3x-29\right)\left(x-8\right)$$

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.

$$3x^{2}-53x+232=0$$

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.

$$x=\frac{-\left(-53\right)±\sqrt{\left(-53\right)^{2}-4\times 3\times 232}}{2\times 3}$$

Square -53.

$$x=\frac{-\left(-53\right)±\sqrt{2809-4\times 3\times 232}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-53\right)±\sqrt{2809-12\times 232}}{2\times 3}$$

Multiply -12 times 232.

$$x=\frac{-\left(-53\right)±\sqrt{2809-2784}}{2\times 3}$$

Add 2809 to -2784.

$$x=\frac{-\left(-53\right)±\sqrt{25}}{2\times 3}$$

Take the square root of 25.

$$x=\frac{-\left(-53\right)±5}{2\times 3}$$

The opposite of -53 is 53.

$$x=\frac{53±5}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{53±5}{6}$$

Now solve the equation x=\frac{53±5}{6} when ± is plus. Add 53 to 5.

$$x=\frac{58}{6}$$

Reduce the fraction \frac{58}{6} to lowest terms by extracting and canceling out 2.

$$x=\frac{29}{3}$$

Now solve the equation x=\frac{53±5}{6} when ± is minus. Subtract 5 from 53.

$$x=\frac{48}{6}$$

Divide 48 by 6.

$$x=8$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{29}{3} for x_{1} and 8 for x_{2}.

$$3x^{2}-53x+232=3\left(x-\frac{29}{3}\right)\left(x-8\right)$$

Subtract \frac{29}{3} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$3x^{2}-53x+232=3\times \frac{3x-29}{3}\left(x-8\right)$$

Cancel out 3, the greatest common factor in 3 and 3.

$$3x^{2}-53x+232=\left(3x-29\right)\left(x-8\right)$$

Evaluate

$$\left(x-8\right)\left(3x-29\right)$$