Solve 5m^2+18m-35

$$5 m ^ { 2 } + 18 m - 35$$

Factor

$$\left(5m-7\right)\left(m+5\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 5m^{2}+am+bm-35. To find a and b, set up a system to be solved.

$$a+b=18 ab=5\left(-35\right)=-175$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -175.

$$-1,175 -5,35 -7,25$$

Calculate the sum for each pair.

$$-1+175=174 -5+35=30 -7+25=18$$

The solution is the pair that gives sum 18.

$$a=-7 b=25$$

Rewrite 5m^{2}+18m-35 as \left(5m^{2}-7m\right)+\left(25m-35\right).

$$\left(5m^{2}-7m\right)+\left(25m-35\right)$$

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

$$m\left(5m-7\right)+5\left(5m-7\right)$$

Factor out common term 5m-7 by using distributive property.

$$\left(5m-7\right)\left(m+5\right)$$

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.

$$5m^{2}+18m-35=0$$

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.

$$m=\frac{-18±\sqrt{18^{2}-4\times 5\left(-35\right)}}{2\times 5}$$

Square 18.

$$m=\frac{-18±\sqrt{324-4\times 5\left(-35\right)}}{2\times 5}$$

Multiply -4 times 5.

$$m=\frac{-18±\sqrt{324-20\left(-35\right)}}{2\times 5}$$

Multiply -20 times -35.

$$m=\frac{-18±\sqrt{324+700}}{2\times 5}$$

Add 324 to 700.

$$m=\frac{-18±\sqrt{1024}}{2\times 5}$$

Take the square root of 1024.

$$m=\frac{-18±32}{2\times 5}$$

Multiply 2 times 5.

$$m=\frac{-18±32}{10}$$

Now solve the equation m=\frac{-18±32}{10} when ± is plus. Add -18 to 32.

$$m=\frac{14}{10}$$

Reduce the fraction \frac{14}{10} to lowest terms by extracting and canceling out 2.

$$m=\frac{7}{5}$$

Now solve the equation m=\frac{-18±32}{10} when ± is minus. Subtract 32 from -18.

$$m=-\frac{50}{10}$$

Divide -50 by 10.

$$m=-5$$

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

$$5m^{2}+18m-35=5\left(m-\frac{7}{5}\right)\left(m-\left(-5\right)\right)$$

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

$$5m^{2}+18m-35=5\left(m-\frac{7}{5}\right)\left(m+5\right)$$

Subtract \frac{7}{5} from m by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$5m^{2}+18m-35=5\times \frac{5m-7}{5}\left(m+5\right)$$

Cancel out 5, the greatest common factor in 5 and 5.

$$5m^{2}+18m-35=\left(5m-7\right)\left(m+5\right)$$

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 5

$$x ^ 2 +\frac{18}{5}x -7 = 0$$

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

$$r + s = -\frac{18}{5} rs = -7$$

Two numbers r and s sum up to -\frac{18}{5} exactly when the average of the two numbers is \frac{1}{2}*-\frac{18}{5} = -\frac{9}{5}. 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>

$$r = -\frac{9}{5} - u s = -\frac{9}{5} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -7

$$(-\frac{9}{5} - u) (-\frac{9}{5} + u) = -7$$

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

$$\frac{81}{25} - u^2 = -7$$

Simplify the expression by subtracting \frac{81}{25} on both sides

$$-u^2 = -7-\frac{81}{25} = -\frac{256}{25}$$

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

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

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

$$r =-\frac{9}{5} - \frac{16}{5} = -5 s = -\frac{9}{5} + \frac{16}{5} = 1.400$$

Evaluate

$$\left(5m-7\right)\left(m+5\right)$$