How to find "a" and "b" from: x^2 - 10x +18 = (x - a)^2 +b?

Answer 1

You "complete the square":

x² - 10x + 18 = 0

Normally, you would start by dividing everything by the leading coefficient (the coefficient of x²). However, that is 1, so there is no need.

Subtract the constant:

x² - 10x = -18

Divide the coefficient of x by 2, and square it. The reason that we do this is to make the left side a perfect square, in the form of x² + 2cx + a². The middle term is "2c" in that format, so divide by 2 and square it to find c² (your constant). Now, you add it to the x² + 2ax format that you have now on the left side to get x² + 2cx + c²:

2c = -10
c = -5
c² = (-5)² = 25
x² - 10x + 25 = 7

Now that you have x and c, you factor the left side into (x+c)²:

(x - 5)² = 7

Put it into (x - a)² + b format:

(x - 5)² = 7
(x - 5)² - 7 = 0

Since we made the original equation equal to 0, and this one equal to 0, we can make them equal to each other:

x² - 10x + 18 = (x - 5)² - 7

The format is (x - a)² + b, so extrapolate the a and b from your answer.

a= 5 (not -5)
b= -7

Answer 2

x^2 - 10x +18 = (x - a)^2 +b

x^2 -10x +25 -25 + 18= (x - a)^2 +b
( x^2 -10x +25 ) - 7 = (x - a)^2 +b
( x - 5 )^2 + ( -7 ) = (x - a)^2 +b

by comparing

a = 5 and b = -7

Answer 3

This looks like completing the square.

We do that by taking half of b and squaring it.

-10/2 = -5
(-5)^2 = 25

(x^2 - 10x + 25) + 18 - 25 =
(x - 5)^2 - 7

a = 5 and b = -7

Answer 4

x² - 10x + 18
= (x² - 10x + 25) - 25 + 18
= (x - 5)² - 7
a = 5
b = - 7

Answer 5

x^2-10x + 18

Using the completing the square method:
ax^2-(b/2)x = ax^2-(b/2)x+(b/2)^2 - (b/2)^2
x^2-10x + (25) + 18 - 25
(x^2-10x+25)^2 + -7
(x-5)^2 + -7

a = 5
b = -7