Q. \(x^2+6x+5\)
Answer
To solve the expression \(x^2+6x+5\), factor it by finding two numbers that multiply to \(5\) and add to \(6\): \(1\) and \(5\).
\[
x^2+6x+5=(x+1)(x+5)
\]
Detailed Explanation
We want to simplify the polynomial \(x^2+6x+5\). A common first step is to factor it completely.
Step 1: Identify the factors.
To factor a quadratic \(x^2+6x+5\), we look for numbers \(a\) and \(b\) such that:
- \(a\cdot b=5\)
- \(a+b=6\)
Step 2: Find the correct numbers.
The number \(5\) factors as \(1\cdot 5\) or \(5\cdot 1\), since these are the only integer pairs that multiply to \(5\).
Check their sums:
- \(1+5=6\)
This matches the needed middle coefficient.
Step 3: Write the factorization.
So the quadratic factors as:
\[
x^2+6x+5=(x+1)(x+5)
\]
Final Answer:
\[
x^2+6x+5=(x+1)(x+5)
\]
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Algebra FAQ
Factor \(x^2+6x+5\) into \((x+a)(x+b)\)?
Compute factors with \(ab=5\) and \(a+b=6\). The pair is \(1\) and \(5\). So \(x^2+6x+5=(x+1)(x+5)\).
Solve \(x^2+6x+5=0\) using factoring?
From \((x+1)(x+5)=0\), set each factor to \(0\). Then \(x=-1\) or \(x=-5\).
Find the vertex form of \(x^2+6x+5\)?
Complete the square: \(x^2+6x+5=(x+3)^2-4\).
What is the discriminant of \(x^2+6x+5\)?
For \(ax^2+bx+c\), \(D=b^2-4ac\). Here \(D=6^2-4(1)(5)=36-20=16\).
Determine the intercepts of \(y=x^2+6x+5\)?
\(y\)-intercept: set \(x=0\), giving \(y=5\). \(x\)-intercepts: solve \(x^2+6x+5=0\) giving \(x=-1\) and \(x=-5\).
What are the roots and their multiplicities?
From factoring, roots are \(x=-1\) and \(x=-5\). Each has multiplicity \(1\) since the factors are linear and distinct.
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