Question

Factor completely.
\[25+70x+49x^2=\]

Answer 1

To factor the expression \(25 + 70x + 49x^2\), we first rearrange it in standard polynomial form:

\[
49x^2 + 70x + 25
\]

Next, we look for two numbers that multiply to \(49 \cdot 25 = 1225\) and add up to \(70\).

Calculating \(49 \cdot 25\):

\[
49 \cdot 25 = 1225
\]

We find the square root of 1225:

\[
\sqrt{1225} = 35
\]

Now, we can express \(70x\) as \(35x + 35x\). Thus, the expression can be written as:

\[
49x^2 + 35x + 35x + 25
\]

Next, we can group the terms:

\[
(49x^2 + 35x) + (35x + 25)
\]

Now, factor out the common terms in each group:

\[
7x(7x + 5) + 5(7x + 5)
\]

Now we notice that we have a common factor of \((7x + 5)\):

\[
(7x + 5)(7x + 5)
\]

This can also be written as:

\[
(7x + 5)^2
\]

Thus, the completely factored form of the expression \(25 + 70x + 49x^2\) is:

\[
\boxed{(7x + 5)^2}
\]