Solve 35m^2+31m+6

$$35 m ^ { 2 } + 31 m + 6$$

Factor

$$\left(5m+3\right)\left(7m+2\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 35m^{2}+am+bm+6. To find a and b, set up a system to be solved.

$$a+b=31 ab=35\times 6=210$$

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 210.

$$1,210 2,105 3,70 5,42 6,35 7,30 10,21 14,15$$

Calculate the sum for each pair.

$$1+210=211 2+105=107 3+70=73 5+42=47 6+35=41 7+30=37 10+21=31 14+15=29$$

The solution is the pair that gives sum 31.

$$a=10 b=21$$

Rewrite 35m^{2}+31m+6 as \left(35m^{2}+10m\right)+\left(21m+6\right).

$$\left(35m^{2}+10m\right)+\left(21m+6\right)$$

Factor out 5m in the first and 3 in the second group.

$$5m\left(7m+2\right)+3\left(7m+2\right)$$

Factor out common term 7m+2 by using distributive property.

$$\left(7m+2\right)\left(5m+3\right)$$

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.

$$35m^{2}+31m+6=0$$

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.

$$m=\frac{-31±\sqrt{31^{2}-4\times 35\times 6}}{2\times 35}$$

Square 31.

$$m=\frac{-31±\sqrt{961-4\times 35\times 6}}{2\times 35}$$

Multiply -4 times 35.

$$m=\frac{-31±\sqrt{961-140\times 6}}{2\times 35}$$

Multiply -140 times 6.

$$m=\frac{-31±\sqrt{961-840}}{2\times 35}$$

Add 961 to -840.

$$m=\frac{-31±\sqrt{121}}{2\times 35}$$

Take the square root of 121.

$$m=\frac{-31±11}{2\times 35}$$

Multiply 2 times 35.

$$m=\frac{-31±11}{70}$$

Now solve the equation m=\frac{-31±11}{70} when ± is plus. Add -31 to 11.

$$m=-\frac{20}{70}$$

Reduce the fraction \frac{-20}{70} to lowest terms by extracting and canceling out 10.

$$m=-\frac{2}{7}$$

Now solve the equation m=\frac{-31±11}{70} when ± is minus. Subtract 11 from -31.

$$m=-\frac{42}{70}$$

Reduce the fraction \frac{-42}{70} to lowest terms by extracting and canceling out 14.

$$m=-\frac{3}{5}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{2}{7} for x_{1} and -\frac{3}{5} for x_{2}.

$$35m^{2}+31m+6=35\left(m-\left(-\frac{2}{7}\right)\right)\left(m-\left(-\frac{3}{5}\right)\right)$$

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

$$35m^{2}+31m+6=35\left(m+\frac{2}{7}\right)\left(m+\frac{3}{5}\right)$$

Add \frac{2}{7} to m by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$35m^{2}+31m+6=35\times \frac{7m+2}{7}\left(m+\frac{3}{5}\right)$$

Add \frac{3}{5} to m by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$35m^{2}+31m+6=35\times \frac{7m+2}{7}\times \frac{5m+3}{5}$$

Multiply \frac{7m+2}{7} times \frac{5m+3}{5} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$35m^{2}+31m+6=35\times \frac{\left(7m+2\right)\left(5m+3\right)}{7\times 5}$$

Multiply 7 times 5.

$$35m^{2}+31m+6=35\times \frac{\left(7m+2\right)\left(5m+3\right)}{35}$$

Cancel out 35, the greatest common factor in 35 and 35.

$$35m^{2}+31m+6=\left(7m+2\right)\left(5m+3\right)$$

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 35

$$x ^ 2 +\frac{31}{35}x +\frac{6}{35} = 0$$

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

$$r + s = -\frac{31}{35} rs = \frac{6}{35}$$

Two numbers r and s sum up to -\frac{31}{35} exactly when the average of the two numbers is \frac{1}{2}*-\frac{31}{35} = -\frac{31}{70}. 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>

$$r = -\frac{31}{70} - u s = -\frac{31}{70} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{6}{35}

$$(-\frac{31}{70} - u) (-\frac{31}{70} + u) = \frac{6}{35}$$

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

$$\frac{961}{4900} - u^2 = \frac{6}{35}$$

Simplify the expression by subtracting \frac{961}{4900} on both sides

$$-u^2 = \frac{6}{35}-\frac{961}{4900} = -\frac{121}{4900}$$

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

$$u^2 = \frac{121}{4900} u = \pm\sqrt{\frac{121}{4900}} = \pm \frac{11}{70}$$

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

$$r =-\frac{31}{70} - \frac{11}{70} = -0.600 s = -\frac{31}{70} + \frac{11}{70} = -0.286$$

Evaluate

$$\left(5m+3\right)\left(7m+2\right)$$