What are the zeroes of f(x) = x2 − 2x − 3? x = −3, 1 x = −3, −1 x = 3, 1 x = 3, −1

1. Break Down the Problem

We need to find the zeroes of the function $f(x) = x^2 - 2x - 3$. This means we need to solve the equation $f(x) = 0$.

2. Relevant Concepts

To find the zeroes of the quadratic function, we can use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a = 1$, $b = -2$, and $c = -3$.

3. Analysis and Detail

Let's apply the quadratic formula step by step:

1. Calculate the discriminant: $b^2 - 4ac = (-2)^2 - 4(1)(-3) = 4 + 12 = 16$

2. Substitute the values into the quadratic formula: $x = \frac{-(-2) \pm \sqrt{16}}{2 \cdot 1} = \frac{2 \pm 4}{2}$

3. Solve for the two possible values of $x$:

   • First solution: $x_1 = \frac{2 + 4}{2} = \frac{6}{2} = 3$
   • Second solution: $x_2 = \frac{2 - 4}{2} = \frac{-2}{2} = -1$

4. Verify and Summarize

The solutions we found are $x = 3$ and $x = -1$. Both values can be verified by substituting back into the original function $f(x)$:

• For $x = 3$: $f(3) = 3^2 - 2(3) - 3 = 9 - 6 - 3 = 0$
• For $x = -1$: $f(-1) = (-1)^2 - 2(-1) - 3 = 1 + 2 - 3 = 0$

Thus, both values are indeed zeroes of the function.

Final Answer

The zeroes of the function $f(x) = x^2 - 2x - 3$ are $x = 3$ and $x = -1$.