# How do you solve #x^2-10x+9=0#?

##### 2 Answers

#### Explanation:

To factor you must play with factors which multiply to

Now, solve each factor for

and

#### Explanation:

Given:

#x^2-10x+9 = 0#

**Sum of coefficients shortcut**

Notice that the sum of the coefficients is

That is:

#1-10+9 = 0#

Hence

#x^2-10x+9 = (x-1)(x-9)#

There are several ways to spot that the other factor must be

So the other solution is

**Completing the square**

Another method, which is a little over the top for this particular problem, involves completing the square, then using the difference of squares identity:

#a^2-b^2 = (a-b)(a+b)#

with

#0 = x^2-10x+9#

#color(white)(0) = x^2-10x+25-16#

#color(white)(0) = (x-5)^2-4^2#

#color(white)(0) = ((x-5)-4)((x-5)+4)#

#color(white)(0) = (x-9)(x-1)#

Hence solutions

**Quadratic formula**

For completeness, I should also mention the quadratic formula.

The equation:

#x^2-10x+9=0#

is in the form

#ax^2+bx+c=0#

with

This has solutions given by the quadratic formula:

#x = (-b+-sqrt(b^2-4ac))/(2a)#

#color(white)(x) = (10+-sqrt((-10)^2-4(1)(9)))/(2*1)#

#color(white)(x) = (10+-sqrt(100-36))/2#

#color(white)(x) = (10+-sqrt(64))/2#

#color(white)(x) = (10+-8)/2#

#color(white)(x) = 5+-4#

That is:

#x = 5+4 = 9" "# or#" "x = 5-4 = 1#