Solve 2x^2+28x+48

$$2 x ^ { 2 } + 28 x + 48$$

Factor

$$2\left(x+2\right)\left(x+12\right)$$

Solution Steps

Factor out 2.

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

Consider x^{2}+14x+24. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+24. To find a and b, set up a system to be solved.

$$a+b=14 ab=1\times 24=24$$

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 14.

$$a=2 b=12$$

Rewrite x^{2}+14x+24 as \left(x^{2}+2x\right)+\left(12x+24\right).

$$\left(x^{2}+2x\right)+\left(12x+24\right)$$

Factor out x in the first and 12 in the second group.

$$x\left(x+2\right)+12\left(x+2\right)$$

Factor out common term x+2 by using distributive property.

$$\left(x+2\right)\left(x+12\right)$$

Rewrite the complete factored expression.

$$2\left(x+2\right)\left(x+12\right)$$

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.

$$2x^{2}+28x+48=0$$

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.

$$x=\frac{-28±\sqrt{28^{2}-4\times 2\times 48}}{2\times 2}$$

Square 28.

$$x=\frac{-28±\sqrt{784-4\times 2\times 48}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-28±\sqrt{784-8\times 48}}{2\times 2}$$

Multiply -8 times 48.

$$x=\frac{-28±\sqrt{784-384}}{2\times 2}$$

Add 784 to -384.

$$x=\frac{-28±\sqrt{400}}{2\times 2}$$

Take the square root of 400.

$$x=\frac{-28±20}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-28±20}{4}$$

Now solve the equation x=\frac{-28±20}{4} when ± is plus. Add -28 to 20.

$$x=-\frac{8}{4}$$

Divide -8 by 4.

$$x=-2$$

Now solve the equation x=\frac{-28±20}{4} when ± is minus. Subtract 20 from -28.

$$x=-\frac{48}{4}$$

Divide -48 by 4.

$$x=-12$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -2 for x_{1} and -12 for x_{2}.

$$2x^{2}+28x+48=2\left(x-\left(-2\right)\right)\left(x-\left(-12\right)\right)$$

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

$$2x^{2}+28x+48=2\left(x+2\right)\left(x+12\right)$$

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

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

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

$$r + s = -14 rs = 24$$

Two numbers r and s sum up to -14 exactly when the average of the two numbers is \frac{1}{2}*-14 = -7. 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>

$$r = -7 - u s = -7 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 24

$$(-7 - u) (-7 + u) = 24$$

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

$$49 - u^2 = 24$$

Simplify the expression by subtracting 49 on both sides

$$-u^2 = 24-49 = -25$$

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

$$u^2 = 25 u = \pm\sqrt{25} = \pm 5$$

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

$$r =-7 - 5 = -12 s = -7 + 5 = -2$$

Evaluate

$$2\left(x+2\right)\left(x+12\right)$$