Q. \(x^2 – 2x – 3\)

Q: Solve x^2-2x-3=0

(x-3)(x+1)=0 \Rightarrow x=3 \text{ or } x=-1.

Q: Find the y-intercept of y=x^2-2x-3

Set x=0. Then y=-3, so the intercept is (0,-3).

Q: Find the vertex of y=x^2-2x-3

x=\frac{-b}{2a}=\frac{2}{2}=1. Then y=1-2-3=-4. Vertex is (1,-4).

Answer

Factor the quadratic.

Find two numbers that multiply to \(-3\) and add to \(-2\): \(-3\) and \(1\).

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

Zeros of the equation

\[
(x-3)(x+1)=0 \;\Rightarrow\; x=3 \text{ or } x=-1
\]

Detailed Explanation

We want to simplify and (most commonly) factor the expression \(x^2 – 2x – 3\).

Step 1: Identify the type of expression

\(x^2 – 2x – 3\) is a quadratic polynomial of the form \(ax^2 + bx + c\), where \(a = 1\), \(b = -2\), and \(c = -3\).

Step 2: Factor the quadratic

We look for two numbers that multiply to \(ac = 1 \cdot (-3) = -3\) and add to \(b = -2\).

The numbers are \(-3\) and \(+1\), because:

\( (-3)(1) = -3 \) and \( (-3) + 1 = -2 \).

Step 3: Write the factorization

Using those numbers, split the middle term \(-2x\) as \(-3x + 1x\).

\[
x^2 – 2x – 3 = x^2 – 3x + x – 3
\]

Step 4: Group and factor

Group terms into two pairs: \((x^2 – 3x)\) and \((x – 3)\).

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

Factor each group:

\[
x^2 – 3x = x(x – 3)
\]

\[
x – 3 = 1(x – 3)
\]

So the expression becomes:

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

Factor out the common factor \((x – 3)\):

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

Final Answer

\[
x^2 – 2x – 3 = (x + 1)(x – 3)
\]

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Algebra FAQ

Factor \(x^2-2x-3\)

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

Solve \(x^2-2x-3=0\)

\( (x-3)(x+1)=0 \Rightarrow x=3 \text{ or } x=-1\).

What are the roots of \(x^2-2x-3\)

The roots are \(x=3\) and \(x=-1\).

Find the \(y\)-intercept of \(y=x^2-2x-3\)

Set \(x=0\). Then \(y=-3\), so the intercept is \((0,-3)\).

Find the vertex of \(y=x^2-2x-3\)

\(x=\frac{-b}{2a}=\frac{2}{2}=1\). Then \(y=1-2-3=-4\). Vertex is \((1,-4)\).

Complete the square for \(x^2-2x-3\)

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

Determine the value of \(x^2-2x-3\) at \(x=2\)

\(4-4-3=-3\).

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