Solve x2 − 2x − 3 = 0

Analysis

The question asks us to find the roots of the quadratic equation $x^2 - 2x - 3 = 0$. This is a standard quadratic equation in the form $ax^2 + bx + c = 0$.

Concept

To solve a quadratic equation, we can use the factorization method. We look for two numbers that:

1. Multiply to give the product of the first and last terms ($1 \times -3 = -3$).
2. Add to give the middle coefficient ($-2$).

Once we find these two numbers, we split the middle term and factor by grouping.

Step-by-Step Solution

Step 1: Identify the factors. We need two numbers that multiply to $-3$ and add to $-2$.

• The factors of $-3$ are $(-3 \times 1)$ or $(3 \times -1)$.
• Check the sums: • $-3 + 1 = -2$ (This matches our middle coefficient!) • $3 + (-1) = 2$ (Incorrect)

So, our numbers are $-3$ and $+1$.

Step 2: Rewrite the equation. Replace $-2x$ with $-3x + 1x$: $x^2 - 3x + 1x - 3 = 0$

Step 3: Factor by grouping. Group the first two terms and the last two terms: $(x^2 - 3x) + (1x - 3) = 0$

Factor out the common terms from each group: $x(x - 3) + 1(x - 3) = 0$

Now, factor out the common binomial $(x - 3)$: $(x - 3)(x + 1) = 0$

Step 4: Solve for x. Set each factor to zero:

• $x - 3 = 0 \implies x = 3$
• $x + 1 = 0 \implies x = -1$

Thus, $x = 3$ or $x = -1$.

Conclusion

Comparing our result with the provided options, we see that the correct values are $3$ and $-1$. This matches option B.

Exam Tip

If you are ever unsure about your answer in the exam hall, plug your results back into the original equation.

• If $x = 3$: $(3)^2 - 2(3) - 3 = 9 - 6 - 3 = 0$. (Correct)
• If $x = -1$: $(-1)^2 - 2(-1) - 3 = 1 + 2 - 3 = 0$. (Correct) This quick check takes less than 30 seconds and guarantees you don't lose marks due to simple sign errors!