Solve 2x^2+14x+24

$$2 x ^ { 2 } + 14 x + 24$$

Factor

$$2\left(x+3\right)\left(x+4\right)$$

Solution Steps

Factor out 2.

$$2\left(x^{2}+7x+12\right)$$

Consider x^{2}+7x+12. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+12. To find a and b, set up a system to be solved.

$$a+b=7 ab=1\times 12=12$$

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 12.

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

Calculate the sum for each pair.

$$1+12=13 2+6=8 3+4=7$$

The solution is the pair that gives sum 7.

$$a=3 b=4$$

Rewrite x^{2}+7x+12 as \left(x^{2}+3x\right)+\left(4x+12\right).

$$\left(x^{2}+3x\right)+\left(4x+12\right)$$

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

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

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

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

Rewrite the complete factored expression.

$$2\left(x+3\right)\left(x+4\right)$$

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.

$$2x^{2}+14x+24=0$$

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.

$$x=\frac{-14±\sqrt{14^{2}-4\times 2\times 24}}{2\times 2}$$

Square 14.

$$x=\frac{-14±\sqrt{196-4\times 2\times 24}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-14±\sqrt{196-8\times 24}}{2\times 2}$$

Multiply -8 times 24.

$$x=\frac{-14±\sqrt{196-192}}{2\times 2}$$

Add 196 to -192.

$$x=\frac{-14±\sqrt{4}}{2\times 2}$$

Take the square root of 4.

$$x=\frac{-14±2}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-14±2}{4}$$

Now solve the equation x=\frac{-14±2}{4} when ± is plus. Add -14 to 2.

$$x=-\frac{12}{4}$$

Divide -12 by 4.

$$x=-3$$

Now solve the equation x=\frac{-14±2}{4} when ± is minus. Subtract 2 from -14.

$$x=-\frac{16}{4}$$

Divide -16 by 4.

$$x=-4$$

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

$$2x^{2}+14x+24=2\left(x-\left(-3\right)\right)\left(x-\left(-4\right)\right)$$

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

$$2x^{2}+14x+24=2\left(x+3\right)\left(x+4\right)$$

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 2

$$x ^ 2 +7x +12 = 0$$

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

$$r + s = -7 rs = 12$$

Two numbers r and s sum up to -7 exactly when the average of the two numbers is \frac{1}{2}*-7 = -\frac{7}{2}. 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>

$$r = -\frac{7}{2} - u s = -\frac{7}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 12

$$(-\frac{7}{2} - u) (-\frac{7}{2} + u) = 12$$

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

$$\frac{49}{4} - u^2 = 12$$

Simplify the expression by subtracting \frac{49}{4} on both sides

$$-u^2 = 12-\frac{49}{4} = -\frac{1}{4}$$

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

$$u^2 = \frac{1}{4} u = \pm\sqrt{\frac{1}{4}} = \pm \frac{1}{2}$$

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

$$r =-\frac{7}{2} - \frac{1}{2} = -4 s = -\frac{7}{2} + \frac{1}{2} = -3$$

Evaluate

$$2\left(x+3\right)\left(x+4\right)$$