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"Write the following expressions in the form $X^2 − A^2$ using the method of completing the square."

"1) $x^2 + 8x + 9$"

I don't understand how that can be, but I did try to do it, with no obvious next steps.

My solution:-

$x^2 + 8x + 16 -16 + 9$

$(x + 4)^2 - 7$

I completed the square, but how can I put that in the form $X^2 - A^2$?

  • 1
    See this video for example for completing the square.2012-03-05

2 Answers 2

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'$x$' isn't the same as '$X$', so $(x+4)^2 − 7$ can be written in the form $X^2 - A^2$. Particularly, $(x+4)$ can be considered as $X$, and $\sqrt{7}$ can be considered as $A$. So, the expression in the designated form would be $(x+4)^2 - (\sqrt{7})^2$.

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In general, given $ax^2 +bx + c\ \forall x \in \mathbb R$ where $a \ne 0$.$ax^2+bx+c= X^2- A^2$ where $X= x+\frac{b}{2a} $ and $A= \frac {\sqrt{b^2-4ac}}{2a}$. Using the quadratic formula and some algebraic mapulation we have the above result