Solving Quadratic Equations by Graphing
The solutions of are exactly where the parabola crosses the x-axis. Those -intercepts are also called the zeros or roots.
So solving by graphing means reading off the -intercepts. The number of crossings tells you how many real solutions there are.
Roots are -intercepts
At an -intercept, , which is exactly the equation you are solving. So each -intercept is a real solution.
If the parabola crosses the axis at and , those two values are the solutions of the related equation.
The parabola below meets the x-axis at two points; reading off those two -intercepts gives the real solutions of the related equation.
Counting solutions from the graph
Two crossings mean two real solutions. A parabola that just touches the axis at its vertex has one repeated solution.
A parabola that never reaches the axis has no real solutions — its solutions are complex. The graph tells you which case you are in at a glance.
Worked examples
Example 1: reading the roots
A parabola crosses the x-axis at and . What are the solutions of the related equation?
Answer: and
Example 2: no real solutions
What does it mean if a parabola never touches the x-axis?
Answer: No real solutions
Example 3: a parabola that only touches the axis
A parabola meets the x-axis at exactly one point, its vertex . What are the solutions?
Answer: , a repeated real solution
Try one yourself
Common questions
Why are -intercepts the solutions?
Solving means finding where — and is exactly the x-axis. So the crossings are the solutions.
What if the parabola only touches the axis once?
That is a double root — one repeated real solution, occurring at the vertex.
Can graphing miss solutions?
It can only show real solutions. If the parabola misses the axis, the solutions are complex and must be found algebraically.
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