Question

How do you solve `x^2-20x+99=0`?

Answer 1

See the solution process below:

Explanation:

Because `-11 - 9 = -20` and `-9 xx -11 = 99` we can factor this equation as:

`(x - 9)(x - 11) = 0`

We can now solve each term for `0` to find the solutions:

Solution 1)

`x - 9 = 0`

`x - 9 + color(red)(9) = 0 + color(red)(9)`

`x - 0 = 9`

`x = 9`

Solution 2)

`x - 11 = 0`

`x - 11 + color(red)(11) = 0 + color(red)(11)`

`x - 0 = 11`

`x = 11`

The solution is: `x = 9` and `x = 11`

Answer 2

`x=9, 11`

Explanation:

There are a number of ways to solve polynomials. We can graph the equation, or use the quadratic formula, or factor. Let's start with factoring and see if we can solve it that way. Then we'll move on to the quadratic formula

`x^2-20x+99`
We are looking for two numbers that combine to `-20` and multiply to `99`.

`color(white)(..)-20`
`color(white)(..)`x`color(white)(.)99`
..............
`color(white)(..)1*99`
`color(white)(..)3*33`
`color(white)(..)color(orange)(9*11)`

`-9+ -11` gives us `-20`, and `-9*-11` equals `99`. So our factors are `(x-9)` and `(x-11)`.

`(x-9)(x-11)=0`
Now we set each factor equal to `0`

Factor 1
`x-9=0`
`x=9`

Factor 2
`x-11=0`
`x=11`

So, we solved the equation without using quadratic formula. If you want to use the quadratic formula, just plug `ax^2+bx+c` into `(-b+-sqrt(b^2-4*a*c))/(2a)`. Both methods will give you the same answer.

Answer 3

9 and 11

Explanation:

`x^2 - 20 x + 99 = 0`
Find 2 real roots knowing sum (-b = 20) and product (c = 99).
They are: 9 and 11. (because sum = 20 and product 99)

NOTE: When a = 1, there is no need to solving by grouping and solving the 2 binomials.