Solve 24x^2+1-11x

$$24 x ^ { 2 } + 1 - 11 x$$

Factor

$$\left(3x-1\right)\left(8x-1\right)$$

Solution Steps

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$24x^{2}-11x+1$$

Factor the expression by grouping. First, the expression needs to be rewritten as 24x^{2}+ax+bx+1. To find a and b, set up a system to be solved.

$$a+b=-11 ab=24\times 1=24$$

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 24.

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

Calculate the sum for each pair.

$$-1-24=-25 -2-12=-14 -3-8=-11 -4-6=-10$$

The solution is the pair that gives sum -11.

$$a=-8 b=-3$$

Rewrite 24x^{2}-11x+1 as \left(24x^{2}-8x\right)+\left(-3x+1\right).

$$\left(24x^{2}-8x\right)+\left(-3x+1\right)$$

Factor out 8x in the first and -1 in the second group.

$$8x\left(3x-1\right)-\left(3x-1\right)$$

Factor out common term 3x-1 by using distributive property.

$$\left(3x-1\right)\left(8x-1\right)$$

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.

$$24x^{2}-11x+1=0$$

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.

$$x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times 24}}{2\times 24}$$

Square -11.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\times 24}}{2\times 24}$$

Multiply -4 times 24.

$$x=\frac{-\left(-11\right)±\sqrt{121-96}}{2\times 24}$$

Add 121 to -96.

$$x=\frac{-\left(-11\right)±\sqrt{25}}{2\times 24}$$

Take the square root of 25.

$$x=\frac{-\left(-11\right)±5}{2\times 24}$$

The opposite of -11 is 11.

$$x=\frac{11±5}{2\times 24}$$

Multiply 2 times 24.

$$x=\frac{11±5}{48}$$

Now solve the equation x=\frac{11±5}{48} when ± is plus. Add 11 to 5.

$$x=\frac{16}{48}$$

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

$$x=\frac{1}{3}$$

Now solve the equation x=\frac{11±5}{48} when ± is minus. Subtract 5 from 11.

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

Reduce the fraction \frac{6}{48} to lowest terms by extracting and canceling out 6.

$$x=\frac{1}{8}$$

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

$$24x^{2}-11x+1=24\left(x-\frac{1}{3}\right)\left(x-\frac{1}{8}\right)$$

Subtract \frac{1}{3} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$24x^{2}-11x+1=24\times \frac{3x-1}{3}\left(x-\frac{1}{8}\right)$$

Subtract \frac{1}{8} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$24x^{2}-11x+1=24\times \frac{3x-1}{3}\times \frac{8x-1}{8}$$

Multiply \frac{3x-1}{3} times \frac{8x-1}{8} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$24x^{2}-11x+1=24\times \frac{\left(3x-1\right)\left(8x-1\right)}{3\times 8}$$

Multiply 3 times 8.

$$24x^{2}-11x+1=24\times \frac{\left(3x-1\right)\left(8x-1\right)}{24}$$

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

$$24x^{2}-11x+1=\left(3x-1\right)\left(8x-1\right)$$

Evaluate

$$\left(3x-1\right)\left(8x-1\right)$$