Solve x^2-10x+9

$${ x }^{ 2 } -10x+9$$

Factor

$$\left(x-9\right)\left(x-1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+9. To find a and b, set up a system to be solved.

$$a+b=-10 ab=1\times 9=9$$

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 9.

$$-1,-9 -3,-3$$

Calculate the sum for each pair.

$$-1-9=-10 -3-3=-6$$

The solution is the pair that gives sum -10.

$$a=-9 b=-1$$

Rewrite x^{2}-10x+9 as \left(x^{2}-9x\right)+\left(-x+9\right).

$$\left(x^{2}-9x\right)+\left(-x+9\right)$$

Factor out x in the first and -1 in the second group.

$$x\left(x-9\right)-\left(x-9\right)$$

Factor out common term x-9 by using distributive property.

$$\left(x-9\right)\left(x-1\right)$$

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.

$$x^{2}-10x+9=0$$

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.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 9}}{2}$$

Square -10.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\times 9}}{2}$$

Multiply -4 times 9.

$$x=\frac{-\left(-10\right)±\sqrt{100-36}}{2}$$

Add 100 to -36.

$$x=\frac{-\left(-10\right)±\sqrt{64}}{2}$$

Take the square root of 64.

$$x=\frac{-\left(-10\right)±8}{2}$$

The opposite of -10 is 10.

$$x=\frac{10±8}{2}$$

Now solve the equation x=\frac{10±8}{2} when ± is plus. Add 10 to 8.

$$x=\frac{18}{2}$$

Divide 18 by 2.

$$x=9$$

Now solve the equation x=\frac{10±8}{2} when ± is minus. Subtract 8 from 10.

$$x=\frac{2}{2}$$

Divide 2 by 2.

$$x=1$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 9 for x_{1} and 1 for x_{2}.

$$x^{2}-10x+9=\left(x-9\right)\left(x-1\right)$$

Evaluate

$$\left(x-9\right)\left(x-1\right)$$