Solve 2m^2+30m+52

$$2 m ^ { 2 } + 30 m + 52$$

Factor

$$2\left(m+2\right)\left(m+13\right)$$

Solution Steps

Factor out 2.

$$2\left(m^{2}+15m+26\right)$$

Consider m^{2}+15m+26. Factor the expression by grouping. First, the expression needs to be rewritten as m^{2}+am+bm+26. To find a and b, set up a system to be solved.

$$a+b=15 ab=1\times 26=26$$

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 26.

$$1,26 2,13$$

Calculate the sum for each pair.

$$1+26=27 2+13=15$$

The solution is the pair that gives sum 15.

$$a=2 b=13$$

Rewrite m^{2}+15m+26 as \left(m^{2}+2m\right)+\left(13m+26\right).

$$\left(m^{2}+2m\right)+\left(13m+26\right)$$

Factor out m in the first and 13 in the second group.

$$m\left(m+2\right)+13\left(m+2\right)$$

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

$$\left(m+2\right)\left(m+13\right)$$

Rewrite the complete factored expression.

$$2\left(m+2\right)\left(m+13\right)$$

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.

$$2m^{2}+30m+52=0$$

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.

$$m=\frac{-30±\sqrt{30^{2}-4\times 2\times 52}}{2\times 2}$$

Square 30.

$$m=\frac{-30±\sqrt{900-4\times 2\times 52}}{2\times 2}$$

Multiply -4 times 2.

$$m=\frac{-30±\sqrt{900-8\times 52}}{2\times 2}$$

Multiply -8 times 52.

$$m=\frac{-30±\sqrt{900-416}}{2\times 2}$$

Add 900 to -416.

$$m=\frac{-30±\sqrt{484}}{2\times 2}$$

Take the square root of 484.

$$m=\frac{-30±22}{2\times 2}$$

Multiply 2 times 2.

$$m=\frac{-30±22}{4}$$

Now solve the equation m=\frac{-30±22}{4} when ± is plus. Add -30 to 22.

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

Divide -8 by 4.

$$m=-2$$

Now solve the equation m=\frac{-30±22}{4} when ± is minus. Subtract 22 from -30.

$$m=-\frac{52}{4}$$

Divide -52 by 4.

$$m=-13$$

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 -13 for x_{2}.

$$2m^{2}+30m+52=2\left(m-\left(-2\right)\right)\left(m-\left(-13\right)\right)$$

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

$$2m^{2}+30m+52=2\left(m+2\right)\left(m+13\right)$$

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

$$x ^ 2 +15x +26 = 0$$

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

$$r + s = -15 rs = 26$$

Two numbers r and s sum up to -15 exactly when the average of the two numbers is \frac{1}{2}*-15 = -\frac{15}{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>

$$r = -\frac{15}{2} - u s = -\frac{15}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 26

$$(-\frac{15}{2} - u) (-\frac{15}{2} + u) = 26$$

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

$$\frac{225}{4} - u^2 = 26$$

Simplify the expression by subtracting \frac{225}{4} on both sides

$$-u^2 = 26-\frac{225}{4} = -\frac{121}{4}$$

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

$$u^2 = \frac{121}{4} u = \pm\sqrt{\frac{121}{4}} = \pm \frac{11}{2}$$

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

$$r =-\frac{15}{2} - \frac{11}{2} = -13 s = -\frac{15}{2} + \frac{11}{2} = -2$$

Evaluate

$$2\left(m+2\right)\left(m+13\right)$$