Solve 21+m-10m^2

$$21 + m - 10 m ^ { 2 }$$

Factor

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

Solution Steps

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-10m^{2}+m+21$$

Factor the expression by grouping. First, the expression needs to be rewritten as -10m^{2}+am+bm+21. To find a and b, set up a system to be solved.

$$a+b=1 ab=-10\times 21=-210$$

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 -210.

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

Calculate the sum for each pair.

$$-1+210=209 -2+105=103 -3+70=67 -5+42=37 -6+35=29 -7+30=23 -10+21=11 -14+15=1$$

The solution is the pair that gives sum 1.

$$a=15 b=-14$$

Rewrite -10m^{2}+m+21 as \left(-10m^{2}+15m\right)+\left(-14m+21\right).

$$\left(-10m^{2}+15m\right)+\left(-14m+21\right)$$

Factor out -5m in the first and -7 in the second group.

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

Factor out common term 2m-3 by using distributive property.

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

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.

$$-10m^{2}+m+21=0$$

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.

$$m=\frac{-1±\sqrt{1^{2}-4\left(-10\right)\times 21}}{2\left(-10\right)}$$

Square 1.

$$m=\frac{-1±\sqrt{1-4\left(-10\right)\times 21}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$m=\frac{-1±\sqrt{1+40\times 21}}{2\left(-10\right)}$$

Multiply 40 times 21.

$$m=\frac{-1±\sqrt{1+840}}{2\left(-10\right)}$$

Add 1 to 840.

$$m=\frac{-1±\sqrt{841}}{2\left(-10\right)}$$

Take the square root of 841.

$$m=\frac{-1±29}{2\left(-10\right)}$$

Multiply 2 times -10.

$$m=\frac{-1±29}{-20}$$

Now solve the equation m=\frac{-1±29}{-20} when ± is plus. Add -1 to 29.

$$m=\frac{28}{-20}$$

Reduce the fraction \frac{28}{-20} to lowest terms by extracting and canceling out 4.

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

Now solve the equation m=\frac{-1±29}{-20} when ± is minus. Subtract 29 from -1.

$$m=-\frac{30}{-20}$$

Reduce the fraction \frac{-30}{-20} to lowest terms by extracting and canceling out 10.

$$m=\frac{3}{2}$$

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 \frac{3}{2} for x_{2}.

$$-10m^{2}+m+21=-10\left(m-\left(-\frac{7}{5}\right)\right)\left(m-\frac{3}{2}\right)$$

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

$$-10m^{2}+m+21=-10\left(m+\frac{7}{5}\right)\left(m-\frac{3}{2}\right)$$

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

$$-10m^{2}+m+21=-10\times \frac{-5m-7}{-5}\left(m-\frac{3}{2}\right)$$

Subtract \frac{3}{2} from m by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$-10m^{2}+m+21=-10\times \frac{-5m-7}{-5}\times \frac{-2m+3}{-2}$$

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

$$-10m^{2}+m+21=-10\times \frac{\left(-5m-7\right)\left(-2m+3\right)}{-5\left(-2\right)}$$

Multiply -5 times -2.

$$-10m^{2}+m+21=-10\times \frac{\left(-5m-7\right)\left(-2m+3\right)}{10}$$

Cancel out 10, the greatest common factor in -10 and 10.

$$-10m^{2}+m+21=-\left(-5m-7\right)\left(-2m+3\right)$$

Evaluate

$$21+m-10m^{2}$$