Solve 3x^2+22x+24

$$3 x ^ { 2 } + 22 x + 24$$

Factor

$$\left(x+6\right)\left(3x+4\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 3x^{2}+ax+bx+24. To find a and b, set up a system to be solved.

$$a+b=22 ab=3\times 24=72$$

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 72.

$$1,72 2,36 3,24 4,18 6,12 8,9$$

Calculate the sum for each pair.

$$1+72=73 2+36=38 3+24=27 4+18=22 6+12=18 8+9=17$$

The solution is the pair that gives sum 22.

$$a=4 b=18$$

Rewrite 3x^{2}+22x+24 as \left(3x^{2}+4x\right)+\left(18x+24\right).

$$\left(3x^{2}+4x\right)+\left(18x+24\right)$$

Factor out x in the first and 6 in the second group.

$$x\left(3x+4\right)+6\left(3x+4\right)$$

Factor out common term 3x+4 by using distributive property.

$$\left(3x+4\right)\left(x+6\right)$$

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.

$$3x^{2}+22x+24=0$$

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.

$$x=\frac{-22±\sqrt{22^{2}-4\times 3\times 24}}{2\times 3}$$

Square 22.

$$x=\frac{-22±\sqrt{484-4\times 3\times 24}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-22±\sqrt{484-12\times 24}}{2\times 3}$$

Multiply -12 times 24.

$$x=\frac{-22±\sqrt{484-288}}{2\times 3}$$

Add 484 to -288.

$$x=\frac{-22±\sqrt{196}}{2\times 3}$$

Take the square root of 196.

$$x=\frac{-22±14}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-22±14}{6}$$

Now solve the equation x=\frac{-22±14}{6} when ± is plus. Add -22 to 14.

$$x=-\frac{8}{6}$$

Reduce the fraction \frac{-8}{6} to lowest terms by extracting and canceling out 2.

$$x=-\frac{4}{3}$$

Now solve the equation x=\frac{-22±14}{6} when ± is minus. Subtract 14 from -22.

$$x=-\frac{36}{6}$$

Divide -36 by 6.

$$x=-6$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{4}{3} for x_{1} and -6 for x_{2}.

$$3x^{2}+22x+24=3\left(x-\left(-\frac{4}{3}\right)\right)\left(x-\left(-6\right)\right)$$

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

$$3x^{2}+22x+24=3\left(x+\frac{4}{3}\right)\left(x+6\right)$$

Add \frac{4}{3} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$3x^{2}+22x+24=3\times \frac{3x+4}{3}\left(x+6\right)$$

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

$$3x^{2}+22x+24=\left(3x+4\right)\left(x+6\right)$$

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 3

$$x ^ 2 +\frac{22}{3}x +8 = 0$$

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

$$r + s = -\frac{22}{3} rs = 8$$

Two numbers r and s sum up to -\frac{22}{3} exactly when the average of the two numbers is \frac{1}{2}*-\frac{22}{3} = -\frac{11}{3}. 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-gzdabgg4ehffg0hf.b01.azurefd.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = -\frac{11}{3} - u s = -\frac{11}{3} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 8

$$(-\frac{11}{3} - u) (-\frac{11}{3} + u) = 8$$

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

$$\frac{121}{9} - u^2 = 8$$

Simplify the expression by subtracting \frac{121}{9} on both sides

$$-u^2 = 8-\frac{121}{9} = -\frac{49}{9}$$

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

$$u^2 = \frac{49}{9} u = \pm\sqrt{\frac{49}{9}} = \pm \frac{7}{3}$$

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

$$r =-\frac{11}{3} - \frac{7}{3} = -6 s = -\frac{11}{3} + \frac{7}{3} = -1.333$$

Evaluate

$$\left(x+6\right)\left(3x+4\right)$$