Question

solve the equation $x^2 - 2x - 3 = 0$ by graphing.
$x = \square$ and $x = \square$

Explanation:

Step1: Identify the quadratic function

The equation corresponds to the function $f(x) = x^2 - 2x - 3$. We need to find its x-intercepts (where $f(x)=0$).

Step2: Find vertex of the parabola

Use vertex formula $x = -\frac{b}{2a}$ for $ax^2+bx+c=0$. Here $a=1, b=-2$:
$x = -\frac{-2}{2(1)} = 1$
Substitute $x=1$ to find y-value:
$f(1) = (1)^2 - 2(1) - 3 = 1 - 2 - 3 = -4$
Vertex is $(1, -4)$.

Step3: Find x-intercepts (roots)

Factor the quadratic:
$x^2 - 2x - 3 = (x-3)(x+1) = 0$
Set each factor to 0:
$x-3=0$ or $x+1=0$

Step4: Verify via graph logic

The parabola opens upward (since $a>0$) with vertex at $(1,-4)$, crossing the x-axis at the solved values.

Answer:

$x = 3$ and $x = -1$