Solve 16m^2+26m+9

$$16 m ^ { 2 } + 26 m + 9$$

Factor

$$\left(2m+1\right)\left(8m+9\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 16m^{2}+am+bm+9. To find a and b, set up a system to be solved.

$$a+b=26 ab=16\times 9=144$$

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 144.

$$1,144 2,72 3,48 4,36 6,24 8,18 9,16 12,12$$

Calculate the sum for each pair.

$$1+144=145 2+72=74 3+48=51 4+36=40 6+24=30 8+18=26 9+16=25 12+12=24$$

The solution is the pair that gives sum 26.

$$a=8 b=18$$

Rewrite 16m^{2}+26m+9 as \left(16m^{2}+8m\right)+\left(18m+9\right).

$$\left(16m^{2}+8m\right)+\left(18m+9\right)$$

Factor out 8m in the first and 9 in the second group.

$$8m\left(2m+1\right)+9\left(2m+1\right)$$

Factor out common term 2m+1 by using distributive property.

$$\left(2m+1\right)\left(8m+9\right)$$

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.

$$16m^{2}+26m+9=0$$

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.

$$m=\frac{-26±\sqrt{26^{2}-4\times 16\times 9}}{2\times 16}$$

Square 26.

$$m=\frac{-26±\sqrt{676-4\times 16\times 9}}{2\times 16}$$

Multiply -4 times 16.

$$m=\frac{-26±\sqrt{676-64\times 9}}{2\times 16}$$

Multiply -64 times 9.

$$m=\frac{-26±\sqrt{676-576}}{2\times 16}$$

Add 676 to -576.

$$m=\frac{-26±\sqrt{100}}{2\times 16}$$

Take the square root of 100.

$$m=\frac{-26±10}{2\times 16}$$

Multiply 2 times 16.

$$m=\frac{-26±10}{32}$$

Now solve the equation m=\frac{-26±10}{32} when ± is plus. Add -26 to 10.

$$m=-\frac{16}{32}$$

Reduce the fraction \frac{-16}{32} to lowest terms by extracting and canceling out 16.

$$m=-\frac{1}{2}$$

Now solve the equation m=\frac{-26±10}{32} when ± is minus. Subtract 10 from -26.

$$m=-\frac{36}{32}$$

Reduce the fraction \frac{-36}{32} to lowest terms by extracting and canceling out 4.

$$m=-\frac{9}{8}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{1}{2} for x_{1} and -\frac{9}{8} for x_{2}.

$$16m^{2}+26m+9=16\left(m-\left(-\frac{1}{2}\right)\right)\left(m-\left(-\frac{9}{8}\right)\right)$$

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

$$16m^{2}+26m+9=16\left(m+\frac{1}{2}\right)\left(m+\frac{9}{8}\right)$$

Add \frac{1}{2} to m by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$16m^{2}+26m+9=16\times \frac{2m+1}{2}\left(m+\frac{9}{8}\right)$$

Add \frac{9}{8} to m by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$16m^{2}+26m+9=16\times \frac{2m+1}{2}\times \frac{8m+9}{8}$$

Multiply \frac{2m+1}{2} times \frac{8m+9}{8} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$16m^{2}+26m+9=16\times \frac{\left(2m+1\right)\left(8m+9\right)}{2\times 8}$$

Multiply 2 times 8.

$$16m^{2}+26m+9=16\times \frac{\left(2m+1\right)\left(8m+9\right)}{16}$$

Cancel out 16, the greatest common factor in 16 and 16.

$$16m^{2}+26m+9=\left(2m+1\right)\left(8m+9\right)$$

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 16

$$x ^ 2 +\frac{13}{8}x +\frac{9}{16} = 0$$

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

$$r + s = -\frac{13}{8} rs = \frac{9}{16}$$

Two numbers r and s sum up to -\frac{13}{8} exactly when the average of the two numbers is \frac{1}{2}*-\frac{13}{8} = -\frac{13}{16}. 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>

$$r = -\frac{13}{16} - u s = -\frac{13}{16} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{9}{16}

$$(-\frac{13}{16} - u) (-\frac{13}{16} + u) = \frac{9}{16}$$

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

$$\frac{169}{256} - u^2 = \frac{9}{16}$$

Simplify the expression by subtracting \frac{169}{256} on both sides

$$-u^2 = \frac{9}{16}-\frac{169}{256} = -\frac{25}{256}$$

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

$$u^2 = \frac{25}{256} u = \pm\sqrt{\frac{25}{256}} = \pm \frac{5}{16}$$

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

$$r =-\frac{13}{16} - \frac{5}{16} = -1.125 s = -\frac{13}{16} + \frac{5}{16} = -0.500$$

Evaluate

$$\left(2m+1\right)\left(8m+9\right)$$