Solve 22x-21-5x^2

$$22 x - 21 - 5 x ^ { 2 }$$

Factor

$$\left(3-x\right)\left(5x-7\right)$$

Solution Steps

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

$$-5x^{2}+22x-21$$

Factor the expression by grouping. First, the expression needs to be rewritten as -5x^{2}+ax+bx-21. To find a and b, set up a system to be solved.

$$a+b=22 ab=-5\left(-21\right)=105$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 105.

$$1,105 3,35 5,21 7,15$$

Calculate the sum for each pair.

$$1+105=106 3+35=38 5+21=26 7+15=22$$

The solution is the pair that gives sum 22.

$$a=15 b=7$$

Rewrite -5x^{2}+22x-21 as \left(-5x^{2}+15x\right)+\left(7x-21\right).

$$\left(-5x^{2}+15x\right)+\left(7x-21\right)$$

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

$$5x\left(-x+3\right)-7\left(-x+3\right)$$

Factor out common term -x+3 by using distributive property.

$$\left(-x+3\right)\left(5x-7\right)$$

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.

$$-5x^{2}+22x-21=0$$

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.

$$x=\frac{-22±\sqrt{22^{2}-4\left(-5\right)\left(-21\right)}}{2\left(-5\right)}$$

Square 22.

$$x=\frac{-22±\sqrt{484-4\left(-5\right)\left(-21\right)}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$x=\frac{-22±\sqrt{484+20\left(-21\right)}}{2\left(-5\right)}$$

Multiply 20 times -21.

$$x=\frac{-22±\sqrt{484-420}}{2\left(-5\right)}$$

Add 484 to -420.

$$x=\frac{-22±\sqrt{64}}{2\left(-5\right)}$$

Take the square root of 64.

$$x=\frac{-22±8}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{-22±8}{-10}$$

Now solve the equation x=\frac{-22±8}{-10} when ± is plus. Add -22 to 8.

$$x=-\frac{14}{-10}$$

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

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

Now solve the equation x=\frac{-22±8}{-10} when ± is minus. Subtract 8 from -22.

$$x=-\frac{30}{-10}$$

Divide -30 by -10.

$$x=3$$

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 3 for x_{2}.

$$-5x^{2}+22x-21=-5\left(x-\frac{7}{5}\right)\left(x-3\right)$$

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

$$-5x^{2}+22x-21=-5\times \frac{-5x+7}{-5}\left(x-3\right)$$

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

$$-5x^{2}+22x-21=\left(-5x+7\right)\left(x-3\right)$$

Evaluate

$$\left(3-x\right)\left(5x-7\right)$$