How do you write $4(x +1)^2 + 1$ in the form $(ax +b)^2 + c\;$?

Question

We can write $4x^2+8x+5$ in the form $a(x+b)^2+c$ as $4(x+1)^2+1$. However, the question I am doing asks me to write it in the form $(ax+b)^2+c$. How do I change it to that form?

Answer 1

You can also solve the equation $(ax+b)^2+c\equiv 4x^2+8x+5$ (the $\equiv$ sign means "true for all values of $x$"—the coefficients on both sides of the equation have to be the same). If we expand the left-hand side, we get $$ (ax+b)^2+c\equiv a^2x^2+2abx+(b^2+c)\equiv 4x^2+8x+5 \, . $$ This means that $a^2=4$ and $2ab=8$ and $b^2+c=5$. If $a^2=4$ then $a=2$ or $a=-2$. Suppose that $a=2$. Then, $4b=8$, and so $b=2$. Finally, $4+c=5$, and so $c=1$. This gives us the solution $$ (2x+2)^2+1\equiv 4x^2+8x+5 \, . $$ If $a=-2$, then $-4b=8$, and so $b=-2$. Finally, $4+c=5$, and so $c=1$. This gives us the solution $$ (-2x-2)^2+1 \equiv 4x^2+8x+5 \, . $$ However, the first solution is arguably simpler because it doesn't have any minus signs in it.

Answer 2

If one has already found $a(x+b)^2+c$ form, then provided $a>0$ one can start by re-writing $a$ as $[\sqrt{a}]^2$ and use that $u^2v^2=(uv)^2$ to get to $$ (\sqrt{a}\ x+\sqrt{a}\ b)^2+c.$$ This looks nicest when, as in your example, the original $a$ is a square, so there is no need for the "unsightly" square root symbol. So $(2x+2)^2+1.$

Answer 3

Hint: $$4(x + 1)^2 = \bigg(2(x + 1)\bigg)^2$$ where I have used the identity $u^2 v^2 = (uv)^2$ as coffeemath has noted.

Can you finish?