Solve 7m^2+27m+18

$$7 m ^ { 2 } + 27 m + 18$$

Factor

$$\left(m+3\right)\left(7m+6\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 7m^{2}+am+bm+18. To find a and b, set up a system to be solved.

$$a+b=27 ab=7\times 18=126$$

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 126.

$$1,126 2,63 3,42 6,21 7,18 9,14$$

Calculate the sum for each pair.

$$1+126=127 2+63=65 3+42=45 6+21=27 7+18=25 9+14=23$$

The solution is the pair that gives sum 27.

$$a=6 b=21$$

Rewrite 7m^{2}+27m+18 as \left(7m^{2}+6m\right)+\left(21m+18\right).

$$\left(7m^{2}+6m\right)+\left(21m+18\right)$$

Factor out m in the first and 3 in the second group.

$$m\left(7m+6\right)+3\left(7m+6\right)$$

Factor out common term 7m+6 by using distributive property.

$$\left(7m+6\right)\left(m+3\right)$$

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.

$$7m^{2}+27m+18=0$$

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.

$$m=\frac{-27±\sqrt{27^{2}-4\times 7\times 18}}{2\times 7}$$

Square 27.

$$m=\frac{-27±\sqrt{729-4\times 7\times 18}}{2\times 7}$$

Multiply -4 times 7.

$$m=\frac{-27±\sqrt{729-28\times 18}}{2\times 7}$$

Multiply -28 times 18.

$$m=\frac{-27±\sqrt{729-504}}{2\times 7}$$

Add 729 to -504.

$$m=\frac{-27±\sqrt{225}}{2\times 7}$$

Take the square root of 225.

$$m=\frac{-27±15}{2\times 7}$$

Multiply 2 times 7.

$$m=\frac{-27±15}{14}$$

Now solve the equation m=\frac{-27±15}{14} when ± is plus. Add -27 to 15.

$$m=-\frac{12}{14}$$

Reduce the fraction \frac{-12}{14} to lowest terms by extracting and canceling out 2.

$$m=-\frac{6}{7}$$

Now solve the equation m=\frac{-27±15}{14} when ± is minus. Subtract 15 from -27.

$$m=-\frac{42}{14}$$

Divide -42 by 14.

$$m=-3$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{6}{7} for x_{1} and -3 for x_{2}.

$$7m^{2}+27m+18=7\left(m-\left(-\frac{6}{7}\right)\right)\left(m-\left(-3\right)\right)$$

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

$$7m^{2}+27m+18=7\left(m+\frac{6}{7}\right)\left(m+3\right)$$

Add \frac{6}{7} to m by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$7m^{2}+27m+18=7\times \frac{7m+6}{7}\left(m+3\right)$$

Cancel out 7, the greatest common factor in 7 and 7.

$$7m^{2}+27m+18=\left(7m+6\right)\left(m+3\right)$$

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 7

$$x ^ 2 +\frac{27}{7}x +\frac{18}{7} = 0$$

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

$$r + s = -\frac{27}{7} rs = \frac{18}{7}$$

Two numbers r and s sum up to -\frac{27}{7} exactly when the average of the two numbers is \frac{1}{2}*-\frac{27}{7} = -\frac{27}{14}. 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>

$$r = -\frac{27}{14} - u s = -\frac{27}{14} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{18}{7}

$$(-\frac{27}{14} - u) (-\frac{27}{14} + u) = \frac{18}{7}$$

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

$$\frac{729}{196} - u^2 = \frac{18}{7}$$

Simplify the expression by subtracting \frac{729}{196} on both sides

$$-u^2 = \frac{18}{7}-\frac{729}{196} = -\frac{225}{196}$$

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

$$u^2 = \frac{225}{196} u = \pm\sqrt{\frac{225}{196}} = \pm \frac{15}{14}$$

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

$$r =-\frac{27}{14} - \frac{15}{14} = -3 s = -\frac{27}{14} + \frac{15}{14} = -0.857$$

Evaluate

$$\left(m+3\right)\left(7m+6\right)$$