Q. \(x^2 + 3x + 2\)

Q: Find the zeros of x^2+3x+2?\n

Set (x+1)(x+2)=0, so x=-1 or x=-2.

Q: Solve x^2+3x+2=0 using the quadratic formula?\n

For a=1,b=3,c=2, x=\frac{-3\pm\sqrt{9-8}}{2}=\frac{-3\pm 1}{2}, giving x=-1,-2.

Q: Complete the square for x^2+3x+2?\n

x^2+3x+2=\left(x+\frac{3}{2}\right)^2-\frac{1}{4}.

Q: What is the vertex (minimum point) of y=x^2+3x+2?\n

The vertex is at x=-\frac{b}{2a}=-\frac{3}{2}. Then y=\left(-\frac{3}{2}\right)^2+3\left(-\frac{3}{2}\right)+2=\frac{-1}{4}.

Q: Determine the sign of x^2+3x+2 across real numbers?\n

Since it factors as (x+1)(x+2) and the parabola opens up, the expression is positive for x -1, and negative between -2 and -1.

Answer

To factor the expression \(x^2+3x+2\), find two numbers that multiply to \(2\) and add to \(3\). Those numbers are \(1\) and \(2\).

\[
x^2+3x+2=(x+1)(x+2)
\]

Detailed Explanation

We want to simplify (or possibly factor) the expression \(x^2 + 3x + 2\).

Step 1: Identify the type of expression.
The expression \(x^2 + 3x + 2\) is a quadratic polynomial. Quadratics can often be factored into the form \((x + a)(x + b)\).

Step 2: Use the factoring approach.
We look for two numbers \(a\) and \(b\) such that the product is the constant term and the sum is the coefficient of \(x\).

For \(x^2 + 3x + 2\):
The coefficient of \(x^2\) is \(1\), so we can try \((x + a)(x + b)\).

Step 3: Match the constant term.
In \((x + a)(x + b)\), the constant term is \(ab\).
We need \(ab = 2\).

The factor pairs of \(2\) are:
\(1 \cdot 2 = 2\)
\( (-1) \cdot (-2) = 2\)

Step 4: Match the middle term coefficient.
In \((x + a)(x + b)\), the coefficient of \(x\) is \(a + b\).
We need \(a + b = 3\).

Check the pair \(1\) and \(2\):
\(1 + 2 = 3\), which matches.

Step 5: Write the factored form.
So the quadratic factors as:

\[
x^2 + 3x + 2 = (x + 1)(x + 2)
\]

Final Answer:
\((x + 1)(x + 2)\)

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Algebra FAQ

Factor \(x^2+3x+2\)?\n

\(x^2+3x+2=(x+1)(x+2)\).

Find the zeros of \(x^2+3x+2\)?\n

Set \((x+1)(x+2)=0\), so \(x=-1\) or \(x=-2\).

Solve \(x^2+3x+2=0\) using the quadratic formula?\n

For \(a=1,b=3,c=2\), \(x=\frac{-3\pm\sqrt{9-8}}{2}=\frac{-3\pm 1}{2}\), giving \(x=-1,-2\).

Complete the square for \(x^2+3x+2\)?\n

\(x^2+3x+2=\left(x+\frac{3}{2}\right)^2-\frac{1}{4}\).

What is the vertex (minimum point) of \(y=x^2+3x+2\)?\n

The vertex is at \(x=-\frac{b}{2a}=-\frac{3}{2}\). Then \(y=\left(-\frac{3}{2}\right)^2+3\left(-\frac{3}{2}\right)+2=\frac{-1}{4}\).

Determine the sign of \(x^2+3x+2\) across real numbers?\n

Since it factors as \((x+1)(x+2)\) and the parabola opens up, the expression is positive for \(x<-2\) and \(x>-1\), and negative between \(-2\) and \(-1\).

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