Algebra
Quadratic equations and functions
Solving quadratic equations by graphing
20 practice questions
2 video lessons
Theory + worked examples
Theory
To solve a quadratic by graphing, graph the parabola and read its \(x\)-intercepts — these are the solutions of \(ax^2+bx+c=0\).
Zero, one, or two \(x\)-intercepts match zero, one, or two real solutions.
The \(x\)-intercepts are the solutions.
Solving by graphing.
The connection:
\[ax^2+bx+c=0\ \Leftrightarrow\ x\text{-intercepts}\]
Where the parabola meets the \(x\)-axis gives the solutions.
How to solve by graphing
- Graph the parabola.
- Find where it crosses the \(x\)-axis.
- Read those \(x\)-values as the solutions.
- Note if there are 0, 1, or 2.
Example 1 — Read the roots
Solve \(x^2-4=0\) by graphing.
Solution
The parabola crosses at \(\pm2\).
| \(x\) | \(=\) | \(-2\ \text{and}\ 2\) |
Example 2 — One solution
A parabola touches the \(x\)-axis at \(x=3\). How many solutions?
Solution
A single touch is one repeated solution.
| \(x\) | \(=\) | \(3\ \text{(once)}\) |
Example 3 — No real solution
A parabola stays above the \(x\)-axis. How many real solutions?
Solution
It never crosses, so no real solutions.
Example 4 — Solutions as zeros
What are the solutions called on the graph?
Solution
The zeros or \(x\)-intercepts.
Common pitfalls
The solutions are \(x\)-intercepts, not the vertex.
Graph accurately to read the zeros.
No \(x\)-intercepts means no real solutions.
Frequently asked questions
How do you solve a quadratic by graphing?
Graph it and read the \(x\)-intercepts.
What are the solutions called?
Zeros, roots, or \(x\)-intercepts.
How many solutions can there be?
Zero, one, or two.
What if the parabola doesn't touch the axis?
No real solutions.
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