X squared - x + 5 = (x-a) squared + b. what are the values of a and b
1 answer:
![{x}^{2} - x + 5 = ({x - a})^{2} + b](https://tex.z-dn.net/?f=%20%7Bx%7D%5E%7B2%7D%20%20-%20x%20%2B%205%20%3D%20%20%28%7Bx%20-%20a%7D%29%5E%7B2%7D%20%20%2B%20b)
![{x}^{2} - x + 5 = {x}^{2} + 2ax + {a}^{2} + b](https://tex.z-dn.net/?f=%20%7Bx%7D%5E%7B2%7D%20%20-%20x%20%2B%205%20%3D%20%20%7Bx%7D%5E%7B2%7D%20%20%2B%202ax%20%2B%20%20%7Ba%7D%5E%7B2%7D%20%2B%20b%20)
This is the equation which mean all the terms must be equalls to the like terms over the " = " peer to peer , look :
![{x}^{2} = {x}^{2}](https://tex.z-dn.net/?f=%20%7Bx%7D%5E%7B2%7D%20%20%3D%20%20%7Bx%7D%5E%7B2%7D%20)
And
![- x = 2ax](https://tex.z-dn.net/?f=%20-%20x%20%3D%202ax)
And
![{a}^{2} + b = 5](https://tex.z-dn.net/?f=%20%7Ba%7D%5E%7B2%7D%20%20%2B%20b%20%3D%205)
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Now it's time to find the a and b values ;
![- x = 2ax](https://tex.z-dn.net/?f=%20-%20x%20%3D%202ax)
Divide both sides by x
![\frac{ - x}{x} = \frac{2ax}{x} \\](https://tex.z-dn.net/?f=%20%5Cfrac%7B%20-%20x%7D%7Bx%7D%20%20%3D%20%20%5Cfrac%7B2ax%7D%7Bx%7D%20%20%5C%5C%20)
![- 1 = 2a](https://tex.z-dn.net/?f=%20-%201%20%3D%202a)
Divide both sides by 2
![\frac{ - 1}{2} = \frac{2a}{2} \\](https://tex.z-dn.net/?f=%20%5Cfrac%7B%20-%201%7D%7B2%7D%20%20%3D%20%5Cfrac%7B2a%7D%7B2%7D%20%20%5C%5C%20)
![a = - \frac{1}{2} \\](https://tex.z-dn.net/?f=a%20%3D%20%20-%20%20%5Cfrac%7B1%7D%7B2%7D%20%20%5C%5C%20)
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![{a}^{2} + b = 5](https://tex.z-dn.net/?f=%20%7Ba%7D%5E%7B2%7D%20%20%2B%20b%20%3D%205)
![( { - \frac{ 1}{2} })^{2} + b = 5 \\](https://tex.z-dn.net/?f=%28%20%7B%20%20-%20%5Cfrac%7B%20%201%7D%7B2%7D%20%7D%29%5E%7B2%7D%20%20%2B%20b%20%3D%205%20%5C%5C%20)
![\frac{1}{4} + b = 5 \\](https://tex.z-dn.net/?f=%20%5Cfrac%7B1%7D%7B4%7D%20%20%2B%20b%20%3D%205%20%5C%5C%20)
Subtract both sides ¼
![\frac{1}{4} - \frac{1}{4} + b = 5 - \frac{1}{4} \\](https://tex.z-dn.net/?f=%20%5Cfrac%7B1%7D%7B4%7D%20%20-%20%20%5Cfrac%7B1%7D%7B4%7D%20%20%2B%20b%20%3D%205%20-%20%20%5Cfrac%7B1%7D%7B4%7D%20%20%5C%5C%20)
![b = \frac{20}{4} - \frac{1}{4} \\](https://tex.z-dn.net/?f=b%20%3D%20%20%5Cfrac%7B20%7D%7B4%7D%20%20-%20%20%5Cfrac%7B1%7D%7B4%7D%20%20%5C%5C%20)
![b = \frac{19}{4} \\](https://tex.z-dn.net/?f=b%20%3D%20%20%5Cfrac%7B19%7D%7B4%7D%20%20%5C%5C%20)
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Thus :
![a = - \frac{1}{2} \\](https://tex.z-dn.net/?f=a%20%3D%20%20-%20%20%5Cfrac%7B1%7D%7B2%7D%20%20%5C%5C%20)
![b = \frac{19}{4} = 4.75 \\](https://tex.z-dn.net/?f=b%20%3D%20%20%5Cfrac%7B19%7D%7B4%7D%20%20%3D%204.75%20%5C%5C%20)
There you go...
Have a great day ❤
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