Algebra
Quadratic equations and functions
Quadratic equations
20 practice questions
2 video lessons
Theory + worked examples
Theory
A quadratic equation has degree \(2\):
\[ax^2+bx+c=0,\quad a\neq0.\]
Its graph is a parabola, and it has up to two real solutions (its \(x\)-intercepts).
Put it in standard form (\(=0\)) before solving.
A quadratic graphs as a parabola.
Quadratic equations.
Standard form:
\[ax^2+bx+c=0\]
The solutions are where the parabola crosses the \(x\)-axis.
How to recognize a quadratic
- Check the highest power is \(2\).
- Write it as \(ax^2+bx+c=0\).
- Identify \(a,b,c\).
- Choose a solving method.
Example 1 — Identify a, b, c
For \(x^2-5x+6=0\), give \(a,b,c\).
Solution
Match \(ax^2+bx+c\).
| \(a=1,\ b=-5,\ c=6\) |
Example 2 — Is it quadratic?
Is \(3x+2=0\) quadratic?
Solution
No — the highest power is \(1\), so it is linear.
Example 3 — Set to standard form
Write \(x^2=2x+3\) in standard form.
Solution
Move all terms to one side.
| \(x^2-2x-3\) | \(=\) | \(0\) |
Example 4 — Number of solutions
How many real solutions can a quadratic have?
Solution
Zero, one, or two — where the parabola meets the \(x\)-axis.
Common pitfalls
\(a\neq0\) — otherwise it isn't quadratic.
Set it to \(=0\) before solving.
Up to two solutions, not always two.
Frequently asked questions
What is a quadratic equation?
An equation of the form \(ax^2+bx+c=0\).
What does its graph look like?
A parabola.
How many solutions can it have?
Zero, one, or two real solutions.
Is \(x^2=9\) quadratic?
Yes — it has an \(x^2\) term.
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Solving quadratic equations by graphing
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