Solve 2n^2+22n+56

$$2 n ^ { 2 } + 22 n + 56$$

Factor

$$2\left(n+4\right)\left(n+7\right)$$

Solution Steps

Factor out 2.

$$2\left(n^{2}+11n+28\right)$$

Consider n^{2}+11n+28. Factor the expression by grouping. First, the expression needs to be rewritten as n^{2}+an+bn+28. To find a and b, set up a system to be solved.

$$a+b=11 ab=1\times 28=28$$

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 28.

$$1,28 2,14 4,7$$

Calculate the sum for each pair.

$$1+28=29 2+14=16 4+7=11$$

The solution is the pair that gives sum 11.

$$a=4 b=7$$

Rewrite n^{2}+11n+28 as \left(n^{2}+4n\right)+\left(7n+28\right).

$$\left(n^{2}+4n\right)+\left(7n+28\right)$$

Factor out n in the first and 7 in the second group.

$$n\left(n+4\right)+7\left(n+4\right)$$

Factor out common term n+4 by using distributive property.

$$\left(n+4\right)\left(n+7\right)$$

Rewrite the complete factored expression.

$$2\left(n+4\right)\left(n+7\right)$$

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.

$$2n^{2}+22n+56=0$$

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.

$$n=\frac{-22±\sqrt{22^{2}-4\times 2\times 56}}{2\times 2}$$

Square 22.

$$n=\frac{-22±\sqrt{484-4\times 2\times 56}}{2\times 2}$$

Multiply -4 times 2.

$$n=\frac{-22±\sqrt{484-8\times 56}}{2\times 2}$$

Multiply -8 times 56.

$$n=\frac{-22±\sqrt{484-448}}{2\times 2}$$

Add 484 to -448.

$$n=\frac{-22±\sqrt{36}}{2\times 2}$$

Take the square root of 36.

$$n=\frac{-22±6}{2\times 2}$$

Multiply 2 times 2.

$$n=\frac{-22±6}{4}$$

Now solve the equation n=\frac{-22±6}{4} when ± is plus. Add -22 to 6.

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

Divide -16 by 4.

$$n=-4$$

Now solve the equation n=\frac{-22±6}{4} when ± is minus. Subtract 6 from -22.

$$n=-\frac{28}{4}$$

Divide -28 by 4.

$$n=-7$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -4 for x_{1} and -7 for x_{2}.

$$2n^{2}+22n+56=2\left(n-\left(-4\right)\right)\left(n-\left(-7\right)\right)$$

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

$$2n^{2}+22n+56=2\left(n+4\right)\left(n+7\right)$$

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

$$x ^ 2 +11x +28 = 0$$

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

$$r + s = -11 rs = 28$$

Two numbers r and s sum up to -11 exactly when the average of the two numbers is \frac{1}{2}*-11 = -\frac{11}{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>

$$r = -\frac{11}{2} - u s = -\frac{11}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 28

$$(-\frac{11}{2} - u) (-\frac{11}{2} + u) = 28$$

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

$$\frac{121}{4} - u^2 = 28$$

Simplify the expression by subtracting \frac{121}{4} on both sides

$$-u^2 = 28-\frac{121}{4} = -\frac{9}{4}$$

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

$$u^2 = \frac{9}{4} u = \pm\sqrt{\frac{9}{4}} = \pm \frac{3}{2}$$

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

$$r =-\frac{11}{2} - \frac{3}{2} = -7 s = -\frac{11}{2} + \frac{3}{2} = -4$$

Evaluate

$$2\left(n+4\right)\left(n+7\right)$$