Q. \(x^2+4x+3\)

Q: What are the zeros of x^2+4x+3 ?

Solve (x+1)(x+3)=0: x=-1 or x=-3.

Q: What is the vertex of y=x^2+4x+3 and its value ?

Vertex at x=-2. Value: y=(-2)^2+4(-2)+3=-1.

Q: What is the discriminant of x^2+4x+3 and what does it mean ?

a=1,b=4,c=3. Discriminant \Delta=b^2-4ac=16-12=4>0, so two real roots.

Q: Solve x^2+4x+3=0 using the quadratic formula ?

x=\frac{-4\pm\sqrt{4}}{2}=\frac{-4\pm 2}{2}. Roots: x=-1,-3.

Answer

\(x^2+4x+3\) factors as \(x^2+4x+3=(x+1)(x+3)\). The roots come from setting each factor equal to \(0\).

\[x^2+4x+3=(x+1)(x+3)\]

\[x+1=0 \Rightarrow x=-1,\quad x+3=0 \Rightarrow x=-3\]

Final result: \(x^2+4x+3=(x+1)(x+3)\), and the solutions are \(x=-1\) and \(x=-3\).

Detailed Explanation

We want to simplify and/or factor the quadratic expression \(x^2+4x+3\).

Step 1: Identify the coefficients

Compare the expression with the standard quadratic form \(ax^2+bx+c\).

Here, \(a=1\), \(b=4\), and \(c=3\).

Step 2: Find two numbers that multiply to \(ac\)

We need two numbers whose product is \(a\cdot c = 1\cdot 3 = 3\).

The possible factor pairs for \(3\) are \(1\) and \(3\) (both positive), or \(-1\) and \(-3\) (both negative).

Step 3: Make them add to \(b\)

We also need the same two numbers to add to \(b=4\).

\(1+3=4\), so the correct pair is \(1\) and \(3\).

Step 4: Factor by grouping

Rewrite the middle term \(4x\) as \(x+3x\):

\[
x^2+4x+3 = x^2+x+3x+3.
\]

Now group terms:

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

Factor each group:

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

So the expression becomes:

\[
x^2+4x+3 = x(x+1) + 3(x+1).
\]

Factor out the common binomial \((x+1)\):

\[
x(x+1) + 3(x+1) = (x+3)(x+1).
\]

Final Answer

\[
x^2+4x+3 = (x+1)(x+3).
\]

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Algebra FAQ

What are the factors of \(x^2+4x+3\) ?

Factor: \(x^2+4x+3=(x+1)(x+3)\).

What are the zeros of \(x^2+4x+3\) ?

Solve \((x+1)(x+3)=0\): \(x=-1\) or \(x=-3\).

How do you complete the square for \(x^2+4x+3\) ?

\(x^2+4x+3=(x+2)^2-1\).

What is the vertex of \(y=x^2+4x+3\) and its value ?

Vertex at \(x=-2\). Value: \(y=(-2)^2+4(-2)+3=-1\).

What is the discriminant of \(x^2+4x+3\) and what does it mean ?

\(a=1,b=4,c=3\). Discriminant \(\Delta=b^2-4ac=16-12=4>0\), so two real roots.

Solve \(x^2+4x+3=0\) using the quadratic formula ?

\(x=\frac{-4\pm\sqrt{4}}{2}=\frac{-4\pm 2}{2}\). Roots: \(x=-1,-3\).

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