Question

X2 + 4x – 9

Can someone give me step by explanation and write the equation in vertex form?

Answer 1

`\underline{\,\underline{\bold{(x+2)^2-13}}\,}`

Explanation:

You can do it by completing the square:

`x^2+4x-9\\\\\underbrace{x^2+4x+4}-4-9`

We add 4 to complete the square but we also subtract 4 because we don't want to change the equation (+4-4=0 so it doesn't change anything in equation)

`x^2+4x+4=x^2+2\cdot x\cdot2+2^2`

so we get:

`\underbrace{x^2+4x+4}-4-9\\\\{}\quad (x+2)^2-13`

The vertex is: (-2, -13)

(because x+2=x-h ⇒h=-2 and +q=-13 ⇒ q=-13)

We can also use that:

`ax^2+bx+c=a(x-h)^2+k` for
`h=(-b)/(2a)\,,\quad k=ah^2+bh+c`

`x^2+4x-9\quad\implies\quad a=1\,,\ b=4\\\\h=(-4)/(2\cdot1)=-\frac42=-2\\\\k=(-2)^2+4(-2)-9=4-8-9=-13\\\\a(x-h)^2+k\\\\1(x-(-2))^2+(-13)\\\\(x+2)^2-13`