Solve 5x^2-22x+21

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

Factor

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

Solution Steps

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.

$$a+b=-22 ab=5\times 21=105$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. 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.

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

Square -22.

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

Multiply -4 times 5.

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

Multiply -20 times 21.

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

Add 484 to -420.

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

Take the square root of 64.

$$x=\frac{-\left(-22\right)±8}{2\times 5}$$

The opposite of -22 is 22.

$$x=\frac{22±8}{2\times 5}$$

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.

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

Divide 30 by 10.

$$x=3$$

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

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

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

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

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 3 for x_{1} and \frac{7}{5} for x_{2}.

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

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

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

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

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

Evaluate

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