Q. \(x^2+10x+25\)
Answer
We factor the quadratic \(x^2+10x+25\) as a perfect square.
\[
x^2+10x+25 = (x+5)^2
\]
Final result: \((x+5)^2\)
Detailed Explanation
We want to evaluate and simplify the expression \(x^2 + 10x + 25\).
Step 1: Recognize a perfect square pattern
The expression looks like the expanded form of a square: \((x+a)^2 = x^2 + 2ax + a^2\).
Compare coefficients with \(x^2 + 10x + 25\):
\(x^2\) matches \(x^2\).
\(10x\) must match \(2ax\), so \(2a = 10\).
Solve for \(a\):
\[
2a = 10
\]
\[
a = 5
\]
Step 2: Check the constant term
If \(a = 5\), then \(a^2 = 5^2 = 25\).
This matches the constant term in the expression, \(25\).
Step 3: Write the expression as a square
Since everything matches, we can rewrite:
\[
x^2 + 10x + 25 = (x+5)^2
\]
Final Answer
\[
x^2 + 10x + 25 = (x+5)^2
\]
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Algebra FAQ
How do you factor \(x^2+10x+25\)?
Recognize a perfect square: \(x^2+10x+25=(x+5)^2\).
How do you write \(x^2+10x+25\) in vertex form?
Complete the square: \(x^2+10x+25=(x+5)^2= x^2+10x+25\), so vertex form is \((x+5)^2\).
Solve \(x^2+10x+25=0\).
Use the factored form: \((x+5)^2=0\). Then \(x=-5\) (a double root).
What is the minimum value of \(x^2+10x+25\)?
Since \((x+5)^2 \ge 0\), the minimum is \(0\), occurring at \(x=-5\).
What is the axis of symmetry for \(y=x^2+10x+25\)?
For \(y=(x+5)^2\), the axis is \(x=-5\).
Expand \((x+5)^2\) to check equality.
\((x+5)^2=x^2+10x+25\). This matches the given expression.
Solve x²+10x+25 step by step.
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