Solve 1+11x+24x^2=

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

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 positive, a and b are both positive. List all such integer pairs that give product 24.

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

Calculate the sum for each pair.

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

The solution is the pair that gives sum 11.

$$a=3 b=8$$

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

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

Factor out 3x in 24x^{2}+3x.

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

Factor out common term 8x+1 by using distributive property.

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

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.

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

Square 11.

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

Multiply -4 times 24.

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

Add 121 to -96.

$$x=\frac{-11±\sqrt{25}}{2\times 24}$$

Take the square root of 25.

$$x=\frac{-11±5}{2\times 24}$$

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Multiply 8 times 3.

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

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

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

Evaluate

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