Solve 90m^2-137m-45

$$90 m ^ { 2 } - 137 m - 45$$

Factor

$$\left(5m-9\right)\left(18m+5\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 90m^{2}+am+bm-45. To find a and b, set up a system to be solved.

$$a+b=-137 ab=90\left(-45\right)=-4050$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -4050.

$$1,-4050 2,-2025 3,-1350 5,-810 6,-675 9,-450 10,-405 15,-270 18,-225 25,-162 27,-150 30,-135 45,-90 50,-81 54,-75$$

Calculate the sum for each pair.

$$1-4050=-4049 2-2025=-2023 3-1350=-1347 5-810=-805 6-675=-669 9-450=-441 10-405=-395 15-270=-255 18-225=-207 25-162=-137 27-150=-123 30-135=-105 45-90=-45 50-81=-31 54-75=-21$$

The solution is the pair that gives sum -137.

$$a=-162 b=25$$

Rewrite 90m^{2}-137m-45 as \left(90m^{2}-162m\right)+\left(25m-45\right).

$$\left(90m^{2}-162m\right)+\left(25m-45\right)$$

Factor out 18m in the first and 5 in the second group.

$$18m\left(5m-9\right)+5\left(5m-9\right)$$

Factor out common term 5m-9 by using distributive property.

$$\left(5m-9\right)\left(18m+5\right)$$

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.

$$90m^{2}-137m-45=0$$

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.

$$m=\frac{-\left(-137\right)±\sqrt{\left(-137\right)^{2}-4\times 90\left(-45\right)}}{2\times 90}$$

Square -137.

$$m=\frac{-\left(-137\right)±\sqrt{18769-4\times 90\left(-45\right)}}{2\times 90}$$

Multiply -4 times 90.

$$m=\frac{-\left(-137\right)±\sqrt{18769-360\left(-45\right)}}{2\times 90}$$

Multiply -360 times -45.

$$m=\frac{-\left(-137\right)±\sqrt{18769+16200}}{2\times 90}$$

Add 18769 to 16200.

$$m=\frac{-\left(-137\right)±\sqrt{34969}}{2\times 90}$$

Take the square root of 34969.

$$m=\frac{-\left(-137\right)±187}{2\times 90}$$

The opposite of -137 is 137.

$$m=\frac{137±187}{2\times 90}$$

Multiply 2 times 90.

$$m=\frac{137±187}{180}$$

Now solve the equation m=\frac{137±187}{180} when ± is plus. Add 137 to 187.

$$m=\frac{324}{180}$$

Reduce the fraction \frac{324}{180} to lowest terms by extracting and canceling out 36.

$$m=\frac{9}{5}$$

Now solve the equation m=\frac{137±187}{180} when ± is minus. Subtract 187 from 137.

$$m=-\frac{50}{180}$$

Reduce the fraction \frac{-50}{180} to lowest terms by extracting and canceling out 10.

$$m=-\frac{5}{18}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{9}{5} for x_{1} and -\frac{5}{18} for x_{2}.

$$90m^{2}-137m-45=90\left(m-\frac{9}{5}\right)\left(m-\left(-\frac{5}{18}\right)\right)$$

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

$$90m^{2}-137m-45=90\left(m-\frac{9}{5}\right)\left(m+\frac{5}{18}\right)$$

Subtract \frac{9}{5} from m by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$90m^{2}-137m-45=90\times \frac{5m-9}{5}\left(m+\frac{5}{18}\right)$$

Add \frac{5}{18} to m by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$90m^{2}-137m-45=90\times \frac{5m-9}{5}\times \frac{18m+5}{18}$$

Multiply \frac{5m-9}{5} times \frac{18m+5}{18} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$90m^{2}-137m-45=90\times \frac{\left(5m-9\right)\left(18m+5\right)}{5\times 18}$$

Multiply 5 times 18.

$$90m^{2}-137m-45=90\times \frac{\left(5m-9\right)\left(18m+5\right)}{90}$$

Cancel out 90, the greatest common factor in 90 and 90.

$$90m^{2}-137m-45=\left(5m-9\right)\left(18m+5\right)$$

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 90

$$x ^ 2 -\frac{137}{90}x -\frac{1}{2} = 0$$

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

$$r + s = \frac{137}{90} rs = -\frac{1}{2}$$

Two numbers r and s sum up to \frac{137}{90} exactly when the average of the two numbers is \frac{1}{2}*\frac{137}{90} = \frac{137}{180}. 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>

$$r = \frac{137}{180} - u s = \frac{137}{180} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{1}{2}

$$(\frac{137}{180} - u) (\frac{137}{180} + u) = -\frac{1}{2}$$

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

$$\frac{18769}{32400} - u^2 = -\frac{1}{2}$$

Simplify the expression by subtracting \frac{18769}{32400} on both sides

$$-u^2 = -\frac{1}{2}-\frac{18769}{32400} = -\frac{34969}{32400}$$

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

$$u^2 = \frac{34969}{32400} u = \pm\sqrt{\frac{34969}{32400}} = \pm \frac{187}{180}$$

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

$$r =\frac{137}{180} - \frac{187}{180} = -0.278 s = \frac{137}{180} + \frac{187}{180} = 1.800$$

Evaluate

$$\left(5m-9\right)\left(18m+5\right)$$