Solve 42m^2-89m-21

$$42 m ^ { 2 } - 89 m - 21$$

Factor

$$\left(3m-7\right)\left(14m+3\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 42m^{2}+am+bm-21. To find a and b, set up a system to be solved.

$$a+b=-89 ab=42\left(-21\right)=-882$$

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 -882.

$$1,-882 2,-441 3,-294 6,-147 7,-126 9,-98 14,-63 18,-49 21,-42$$

Calculate the sum for each pair.

$$1-882=-881 2-441=-439 3-294=-291 6-147=-141 7-126=-119 9-98=-89 14-63=-49 18-49=-31 21-42=-21$$

The solution is the pair that gives sum -89.

$$a=-98 b=9$$

Rewrite 42m^{2}-89m-21 as \left(42m^{2}-98m\right)+\left(9m-21\right).

$$\left(42m^{2}-98m\right)+\left(9m-21\right)$$

Factor out 14m in the first and 3 in the second group.

$$14m\left(3m-7\right)+3\left(3m-7\right)$$

Factor out common term 3m-7 by using distributive property.

$$\left(3m-7\right)\left(14m+3\right)$$

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.

$$42m^{2}-89m-21=0$$

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.

$$m=\frac{-\left(-89\right)±\sqrt{\left(-89\right)^{2}-4\times 42\left(-21\right)}}{2\times 42}$$

Square -89.

$$m=\frac{-\left(-89\right)±\sqrt{7921-4\times 42\left(-21\right)}}{2\times 42}$$

Multiply -4 times 42.

$$m=\frac{-\left(-89\right)±\sqrt{7921-168\left(-21\right)}}{2\times 42}$$

Multiply -168 times -21.

$$m=\frac{-\left(-89\right)±\sqrt{7921+3528}}{2\times 42}$$

Add 7921 to 3528.

$$m=\frac{-\left(-89\right)±\sqrt{11449}}{2\times 42}$$

Take the square root of 11449.

$$m=\frac{-\left(-89\right)±107}{2\times 42}$$

The opposite of -89 is 89.

$$m=\frac{89±107}{2\times 42}$$

Multiply 2 times 42.

$$m=\frac{89±107}{84}$$

Now solve the equation m=\frac{89±107}{84} when ± is plus. Add 89 to 107.

$$m=\frac{196}{84}$$

Reduce the fraction \frac{196}{84} to lowest terms by extracting and canceling out 28.

$$m=\frac{7}{3}$$

Now solve the equation m=\frac{89±107}{84} when ± is minus. Subtract 107 from 89.

$$m=-\frac{18}{84}$$

Reduce the fraction \frac{-18}{84} to lowest terms by extracting and canceling out 6.

$$m=-\frac{3}{14}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{7}{3} for x_{1} and -\frac{3}{14} for x_{2}.

$$42m^{2}-89m-21=42\left(m-\frac{7}{3}\right)\left(m-\left(-\frac{3}{14}\right)\right)$$

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

$$42m^{2}-89m-21=42\left(m-\frac{7}{3}\right)\left(m+\frac{3}{14}\right)$$

Subtract \frac{7}{3} from m by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$42m^{2}-89m-21=42\times \frac{3m-7}{3}\left(m+\frac{3}{14}\right)$$

Add \frac{3}{14} to m by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$42m^{2}-89m-21=42\times \frac{3m-7}{3}\times \frac{14m+3}{14}$$

Multiply \frac{3m-7}{3} times \frac{14m+3}{14} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$42m^{2}-89m-21=42\times \frac{\left(3m-7\right)\left(14m+3\right)}{3\times 14}$$

Multiply 3 times 14.

$$42m^{2}-89m-21=42\times \frac{\left(3m-7\right)\left(14m+3\right)}{42}$$

Cancel out 42, the greatest common factor in 42 and 42.

$$42m^{2}-89m-21=\left(3m-7\right)\left(14m+3\right)$$

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 42

$$x ^ 2 -\frac{89}{42}x -\frac{1}{2} = 0$$

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

$$r + s = \frac{89}{42} rs = -\frac{1}{2}$$

Two numbers r and s sum up to \frac{89}{42} exactly when the average of the two numbers is \frac{1}{2}*\frac{89}{42} = \frac{89}{84}. 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>

$$r = \frac{89}{84} - u s = \frac{89}{84} + u$$

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

$$(\frac{89}{84} - u) (\frac{89}{84} + u) = -\frac{1}{2}$$

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

$$\frac{7921}{7056} - u^2 = -\frac{1}{2}$$

Simplify the expression by subtracting \frac{7921}{7056} on both sides

$$-u^2 = -\frac{1}{2}-\frac{7921}{7056} = -\frac{11449}{7056}$$

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

$$u^2 = \frac{11449}{7056} u = \pm\sqrt{\frac{11449}{7056}} = \pm \frac{107}{84}$$

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

$$r =\frac{89}{84} - \frac{107}{84} = -0.214 s = \frac{89}{84} + \frac{107}{84} = 2.333$$

Evaluate

$$\left(3m-7\right)\left(14m+3\right)$$