Solve 2n^2+28n+96

$$2 n ^ { 2 } + 28 n + 96$$

Factor

$$2\left(n+6\right)\left(n+8\right)$$

Solution Steps

Factor out 2.

$$2\left(n^{2}+14n+48\right)$$

Consider n^{2}+14n+48. Factor the expression by grouping. First, the expression needs to be rewritten as n^{2}+an+bn+48. To find a and b, set up a system to be solved.

$$a+b=14 ab=1\times 48=48$$

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 48.

$$1,48 2,24 3,16 4,12 6,8$$

Calculate the sum for each pair.

$$1+48=49 2+24=26 3+16=19 4+12=16 6+8=14$$

The solution is the pair that gives sum 14.

$$a=6 b=8$$

Rewrite n^{2}+14n+48 as \left(n^{2}+6n\right)+\left(8n+48\right).

$$\left(n^{2}+6n\right)+\left(8n+48\right)$$

Factor out n in the first and 8 in the second group.

$$n\left(n+6\right)+8\left(n+6\right)$$

Factor out common term n+6 by using distributive property.

$$\left(n+6\right)\left(n+8\right)$$

Rewrite the complete factored expression.

$$2\left(n+6\right)\left(n+8\right)$$

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.

$$2n^{2}+28n+96=0$$

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.

$$n=\frac{-28±\sqrt{28^{2}-4\times 2\times 96}}{2\times 2}$$

Square 28.

$$n=\frac{-28±\sqrt{784-4\times 2\times 96}}{2\times 2}$$

Multiply -4 times 2.

$$n=\frac{-28±\sqrt{784-8\times 96}}{2\times 2}$$

Multiply -8 times 96.

$$n=\frac{-28±\sqrt{784-768}}{2\times 2}$$

Add 784 to -768.

$$n=\frac{-28±\sqrt{16}}{2\times 2}$$

Take the square root of 16.

$$n=\frac{-28±4}{2\times 2}$$

Multiply 2 times 2.

$$n=\frac{-28±4}{4}$$

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

$$n=-\frac{24}{4}$$

Divide -24 by 4.

$$n=-6$$

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

$$n=-\frac{32}{4}$$

Divide -32 by 4.

$$n=-8$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -6 for x_{1} and -8 for x_{2}.

$$2n^{2}+28n+96=2\left(n-\left(-6\right)\right)\left(n-\left(-8\right)\right)$$

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

$$2n^{2}+28n+96=2\left(n+6\right)\left(n+8\right)$$

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

$$x ^ 2 +14x +48 = 0$$

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

$$r + s = -14 rs = 48$$

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 = 48

$$(-7 - u) (-7 + u) = 48$$

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

$$49 - u^2 = 48$$

Simplify the expression by subtracting 49 on both sides

$$-u^2 = 48-49 = -1$$

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

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

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

$$r =-7 - 1 = -8 s = -7 + 1 = -6$$

Evaluate

$$2\left(n+6\right)\left(n+8\right)$$