Find all values of a and b that will make (ax+5)2=16x2+bx+25 true for all x.

Question

Find all values of a and b that will make  (ax+5)2=16x2+bx+25 true for all x.

caoireoilns · Accepted Answer

Step 1
The quadratic formula is not needed here
${(a x + 5)}^{2} & = 16 x^{2} + b x + 25$
$\Rightarrow b = a^{2} x - 16 x + 10 a$
$\Rightarrow {(a x + 5)}^{2} = 16 x^{2} + (a^{2} x - 16 x + 10 a) x + 25$
$= {(a x + 5)}^{2}$
We can see from
${(a x + 5)}^{2} = {(a x + 5)}^{2}$
how all values of x require that
$b = a^{2} x - 16 x + 10 a$
On the other hand, expanding both sides and comparing coefficients,
$a^{2} x^{2} + 10 a x + 25$
$= 16 x^{2} + b x + 25$
these results are easily verified in a spreadsheet. This is also confirmed by
$b = a^{2} x - 16 x + 10 a$
$= 4^{2} x - 16 x + 10 \times 4$
$= 40$

portafilses · Accepted Answer

If its true for all x, then its true for x=1 and x=−1. Evaluate those values of x, then sum the resulting equations and you will be able to solve for a. Then substitute the value of a to solve for b.

Find all values of a and b that will make (ax+5)2=16x2+bx+25

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