Solve 8x^2+33x+4

$$8 { x }^{ 2 } +33x+4$$

Factor

$$\left(x+4\right)\left(8x+1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 8x^{2}+ax+bx+4. To find a and b, set up a system to be solved.

$$a+b=33 ab=8\times 4=32$$

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 32.

$$1,32 2,16 4,8$$

Calculate the sum for each pair.

$$1+32=33 2+16=18 4+8=12$$

The solution is the pair that gives sum 33.

$$a=1 b=32$$

Rewrite 8x^{2}+33x+4 as \left(8x^{2}+x\right)+\left(32x+4\right).

$$\left(8x^{2}+x\right)+\left(32x+4\right)$$

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

$$x\left(8x+1\right)+4\left(8x+1\right)$$

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

$$\left(8x+1\right)\left(x+4\right)$$

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.

$$8x^{2}+33x+4=0$$

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.

$$x=\frac{-33±\sqrt{33^{2}-4\times 8\times 4}}{2\times 8}$$

Square 33.

$$x=\frac{-33±\sqrt{1089-4\times 8\times 4}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-33±\sqrt{1089-32\times 4}}{2\times 8}$$

Multiply -32 times 4.

$$x=\frac{-33±\sqrt{1089-128}}{2\times 8}$$

Add 1089 to -128.

$$x=\frac{-33±\sqrt{961}}{2\times 8}$$

Take the square root of 961.

$$x=\frac{-33±31}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{-33±31}{16}$$

Now solve the equation x=\frac{-33±31}{16} when ± is plus. Add -33 to 31.

$$x=-\frac{2}{16}$$

Reduce the fraction \frac{-2}{16} to lowest terms by extracting and canceling out 2.

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

Now solve the equation x=\frac{-33±31}{16} when ± is minus. Subtract 31 from -33.

$$x=-\frac{64}{16}$$

Divide -64 by 16.

$$x=-4$$

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 -4 for x_{2}.

$$8x^{2}+33x+4=8\left(x-\left(-\frac{1}{8}\right)\right)\left(x-\left(-4\right)\right)$$

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

$$8x^{2}+33x+4=8\left(x+\frac{1}{8}\right)\left(x+4\right)$$

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

$$8x^{2}+33x+4=8\times \frac{8x+1}{8}\left(x+4\right)$$

Cancel out 8, the greatest common factor in 8 and 8.

$$8x^{2}+33x+4=\left(8x+1\right)\left(x+4\right)$$

Evaluate

$$\left(x+4\right)\left(8x+1\right)$$