Solve 21m^2+21m-42

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

Factor

$$21\left(m-1\right)\left(m+2\right)$$

Solution Steps

Factor out 21.

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

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

$$a+b=1 ab=1\left(-2\right)=-2$$

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. The only such pair is the system solution.

$$a=-1 b=2$$

Rewrite m^{2}+m-2 as \left(m^{2}-m\right)+\left(2m-2\right).

$$\left(m^{2}-m\right)+\left(2m-2\right)$$

Factor out m in the first and 2 in the second group.

$$m\left(m-1\right)+2\left(m-1\right)$$

Factor out common term m-1 by using distributive property.

$$\left(m-1\right)\left(m+2\right)$$

Rewrite the complete factored expression.

$$21\left(m-1\right)\left(m+2\right)$$

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.

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

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.

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

Square 21.

$$m=\frac{-21±\sqrt{441-4\times 21\left(-42\right)}}{2\times 21}$$

Multiply -4 times 21.

$$m=\frac{-21±\sqrt{441-84\left(-42\right)}}{2\times 21}$$

Multiply -84 times -42.

$$m=\frac{-21±\sqrt{441+3528}}{2\times 21}$$

Add 441 to 3528.

$$m=\frac{-21±\sqrt{3969}}{2\times 21}$$

Take the square root of 3969.

$$m=\frac{-21±63}{2\times 21}$$

Multiply 2 times 21.

$$m=\frac{-21±63}{42}$$

Now solve the equation m=\frac{-21±63}{42} when ± is plus. Add -21 to 63.

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

Divide 42 by 42.

$$m=1$$

Now solve the equation m=\frac{-21±63}{42} when ± is minus. Subtract 63 from -21.

$$m=-\frac{84}{42}$$

Divide -84 by 42.

$$m=-2$$

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

$$21m^{2}+21m-42=21\left(m-1\right)\left(m-\left(-2\right)\right)$$

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

$$21m^{2}+21m-42=21\left(m-1\right)\left(m+2\right)$$

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 21

$$x ^ 2 +1x -2 = 0$$

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

$$r + s = -1 rs = -2$$

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

$$r = -\frac{1}{2} - u s = -\frac{1}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -2

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

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

$$\frac{1}{4} - u^2 = -2$$

Simplify the expression by subtracting \frac{1}{4} on both sides

$$-u^2 = -2-\frac{1}{4} = -\frac{9}{4}$$

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

$$u^2 = \frac{9}{4} u = \pm\sqrt{\frac{9}{4}} = \pm \frac{3}{2}$$

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

$$r =-\frac{1}{2} - \frac{3}{2} = -2 s = -\frac{1}{2} + \frac{3}{2} = 1$$

Evaluate

$$21\left(m-1\right)\left(m+2\right)$$