Solve 4x^2+23x+33

$$4 x ^ { 2 } + 23 x + 33$$

Factor

$$\left(x+3\right)\left(4x+11\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 4x^{2}+ax+bx+33. To find a and b, set up a system to be solved.

$$a+b=23 ab=4\times 33=132$$

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 132.

$$1,132 2,66 3,44 4,33 6,22 11,12$$

Calculate the sum for each pair.

$$1+132=133 2+66=68 3+44=47 4+33=37 6+22=28 11+12=23$$

The solution is the pair that gives sum 23.

$$a=11 b=12$$

Rewrite 4x^{2}+23x+33 as \left(4x^{2}+11x\right)+\left(12x+33\right).

$$\left(4x^{2}+11x\right)+\left(12x+33\right)$$

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

$$x\left(4x+11\right)+3\left(4x+11\right)$$

Factor out common term 4x+11 by using distributive property.

$$\left(4x+11\right)\left(x+3\right)$$

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.

$$4x^{2}+23x+33=0$$

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.

$$x=\frac{-23±\sqrt{23^{2}-4\times 4\times 33}}{2\times 4}$$

Square 23.

$$x=\frac{-23±\sqrt{529-4\times 4\times 33}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-23±\sqrt{529-16\times 33}}{2\times 4}$$

Multiply -16 times 33.

$$x=\frac{-23±\sqrt{529-528}}{2\times 4}$$

Add 529 to -528.

$$x=\frac{-23±\sqrt{1}}{2\times 4}$$

Take the square root of 1.

$$x=\frac{-23±1}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-23±1}{8}$$

Now solve the equation x=\frac{-23±1}{8} when ± is plus. Add -23 to 1.

$$x=-\frac{22}{8}$$

Reduce the fraction \frac{-22}{8} to lowest terms by extracting and canceling out 2.

$$x=-\frac{11}{4}$$

Now solve the equation x=\frac{-23±1}{8} when ± is minus. Subtract 1 from -23.

$$x=-\frac{24}{8}$$

Divide -24 by 8.

$$x=-3$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{11}{4} for x_{1} and -3 for x_{2}.

$$4x^{2}+23x+33=4\left(x-\left(-\frac{11}{4}\right)\right)\left(x-\left(-3\right)\right)$$

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

$$4x^{2}+23x+33=4\left(x+\frac{11}{4}\right)\left(x+3\right)$$

Add \frac{11}{4} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$4x^{2}+23x+33=4\times \frac{4x+11}{4}\left(x+3\right)$$

Cancel out 4, the greatest common factor in 4 and 4.

$$4x^{2}+23x+33=\left(4x+11\right)\left(x+3\right)$$

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.This is achieved by dividing both sides of the equation by 4

$$x ^ 2 +\frac{23}{4}x +\frac{33}{4} = 0$$

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = -\frac{23}{4} rs = \frac{33}{4}$$

Two numbers r and s sum up to -\frac{23}{4} exactly when the average of the two numbers is \frac{1}{2}*-\frac{23}{4} = -\frac{23}{8}. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = -\frac{23}{8} - u s = -\frac{23}{8} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{33}{4}

$$(-\frac{23}{8} - u) (-\frac{23}{8} + u) = \frac{33}{4}$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$\frac{529}{64} - u^2 = \frac{33}{4}$$

Simplify the expression by subtracting \frac{529}{64} on both sides

$$-u^2 = \frac{33}{4}-\frac{529}{64} = -\frac{1}{64}$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = \frac{1}{64} u = \pm\sqrt{\frac{1}{64}} = \pm \frac{1}{8}$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =-\frac{23}{8} - \frac{1}{8} = -3 s = -\frac{23}{8} + \frac{1}{8} = -2.750$$

Evaluate

$$\left(x+3\right)\left(4x+11\right)$$