Problem
How to find x intercept of a quadratic is a question about where a parabola crosses the x-axis. Consider the quadratic function \(y = x^2 + 2\cdot x - 3\). The task is to find its x-intercepts and interpret them on the graph.
Concept: what an x-intercept means
An x-intercept is a point where the graph meets the x-axis, so the y-coordinate equals \(0\). Therefore, x-intercepts occur at solutions of the equation \(f(x)=0\).
Key rule: For \(y=f(x)\), x-intercepts are found by solving \(f(x)=0\). Each real solution \(r\) gives an intercept \((r,0)\).
Step-by-step method (general quadratic)
For a quadratic \(y = a\cdot x^2 + b\cdot x + c\) with \(a \ne 0\):
- Set \(y=0\): solve \(a\cdot x^2 + b\cdot x + c = 0\).
- Solve the quadratic (factoring when possible, or use the quadratic formula).
- Keep only real solutions for x-intercepts; write each intercept as \((x,0)\).
Worked example: \(y = x^2 + 2\cdot x - 3\)
Step 1: Set \(y=0\)
\[ 0 = x^2 + 2\cdot x - 3 \]
Step 2: Solve the quadratic (factoring)
Factor by finding two numbers whose product is \(-3\) and whose sum is \(2\): \(3\) and \(-1\).
\[ x^2 + 2\cdot x - 3 = (x + 3)\cdot(x - 1) \]
Set each factor equal to \(0\):
\[ x + 3 = 0 \quad \Rightarrow \quad x = -3 \]
\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \]
Step 3: Write the x-intercepts
x-intercepts: \((-3,0)\) and \((1,0)\)
These are also called the real zeros (real roots) of the quadratic function.
How many x-intercepts can a quadratic have?
The discriminant \(D = b^2 - 4\cdot a\cdot c\) determines the number of real solutions of \(a\cdot x^2 + b\cdot x + c = 0\).
| Discriminant condition | Real solutions for \(x\) | Number of x-intercepts |
|---|---|---|
| \(D > 0\) | Two distinct real roots | Two x-intercepts |
| \(D = 0\) | One repeated real root | One x-intercept (touches the x-axis) |
| \(D < 0\) | No real roots | No x-intercepts |
Visualization: parabola and its x-intercepts
Optional method: quadratic formula (works for every quadratic)
When factoring is difficult or impossible over the integers, the quadratic formula finds the roots of \(a\cdot x^2 + b\cdot x + c = 0\):
\[ x = \frac{-b \pm \sqrt{b^2 - 4\cdot a\cdot c}}{2\cdot a} \]
Real x-intercepts exist only when the expression under the square root is nonnegative: \(b^2 - 4\cdot a\cdot c \ge 0\).