Question

question
type the correct answer in each box.
consider the perfect square trinomial identity:
a² + 2ab + b² = (a + b)².
for the polynomial x² + 10x + 25,
a =
and b =
.

Explanation:

Step1: Identify the form of perfect square

The perfect square trinomial identity is \(a^{2}+2ab + b^{2}=(a + b)^{2}\). For the polynomial \(x^{2}+10x + 25\), we compare it with \(a^{2}+2ab + b^{2}\).
Here, \(a^{2}=x^{2}\), so we solve for \(a\):
\(a=\sqrt{x^{2}}=x\) (assuming we are dealing with real - valued and simple cases, and taking the positive root for the form of the perfect square).

Step2: Find the value of \(b\)

We know that \(2ab = 10x\). Since we found that \(a = x\), we substitute \(a=x\) into \(2ab=10x\).
Substituting \(a = x\) into \(2ab = 10x\), we get \(2\times x\times b=10x\).
Divide both sides of the equation \(2xb = 10x\) by \(2x\) (assuming \(x
eq0\); if \(x = 0\), the polynomial becomes \(25=(0 + b)^{2}\), and \(b = 5\) still holds).
\(b=\frac{10x}{2x}=5\). We can also check by looking at \(b^{2}\). Since \(b^{2}=25\), then \(b=\sqrt{25}=5\) (again, taking the positive root for the form of the perfect square).

Answer:

\(a = x\) and \(b = 5\)