Q. \(x^2 + 8x + 7\)
Answer
We factor the quadratic: \(\,x^2+8x+7\). Find two numbers that multiply to \(7\) and add to \(8\): \(1\) and \(7\). Since \(1\cdot 7=7\) and \(1+7=8\),
\[
x^2+8x+7=(x+1)(x+7).
\]
Detailed Explanation
We want to simplify the expression
\[
x^2+8x+7.
\]
Step 1: Look for a way to factor the quadratic.
For a quadratic of the form
\[
x^2+bx+c,
\]
we try to write it as
\[
(x+m)(x+n),
\]
so that
\[
m+n=b \quad \text{and} \quad mn=c.
\]
Step 2: Identify \(b\) and \(c\).
Here, \(b=8\) and \(c=7\).
Step 3: Find two numbers that multiply to \(7\) and add to \(8\).
List factor pairs of \(7\):
- \(1\) and \(7\) (since \(1\cdot 7=7\))
Check their sum:
\[
1+7=8.
\]
That matches \(b=8\).
Step 4: Write the factored form.
\[
x^2+8x+7=(x+1)(x+7).
\]
Final answer:
\[
\boxed{(x+1)(x+7)}.
\]
See full solution
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Algebra FAQ
What are the roots of \(x^2+8x+7=0\)?
Factor \(x^2+8x+7=(x+1)(x+7)\). So \(x=-1\) or \(x=-7\).
Can \(x^2+8x+7\) be factored?
Yes. Since \(1\cdot 7=7\) and \(1+7=8\), \(x^2+8x+7=(x+1)(x+7)\).
What is the discriminant of \(x^2+8x+7=0\)?
\(a=1,b=8,c=7\). \(D=b^2-4ac=8^2-4\cdot 1\cdot 7=64-28=36\).
What is the completing-the-square form?
\(x^2+8x+7=(x+4)^2-16+7=(x+4)^2-9\).
What is the vertex and minimum value of \(f(x)=x^2+8x+7\)?
Vertex at \(x=-\frac{b}{2a}=-4\). Minimum value \(f(-4)=(-4)^2+8(-4)+7=16-32+7=-9\).
What is the range of \(f(x)=x^2+8x+7\)?
Since \(f(x)=(x+4)^2-9\), \((x+4)^2\ge 0\). So \(f(x)\ge -9\), meaning the range is \([-9,\infty)\).
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