The key ideas
- Factored form reveals zeros; vertex form reveals the maximum or minimum.
- The discriminant tells how many real solutions an equation has.
- A root is also an x-intercept of the related graph.
Start by identifying what the question asks
For roots, rewrite the equation so one side is zero, then factor if the factors are easy to find. The zero-product rule applies when a product equals zero: at least one factor must be zero. If factoring is awkward, the quadratic formula is another route.
For a maximum or minimum, vertex form y = a(x − h)² + k makes the vertex (h, k) visible. For x-intercepts, factored form is often more useful. A question about a function's value is different from a question about its zeros.
Use the discriminant to count real solutions
For ax² + bx + c = 0 with a ≠ 0, calculate b² − 4ac. A positive value means two distinct real roots, zero means one repeated real root, and a negative value means no real roots. You can often answer a question about the number of solutions without solving for the roots.
Worked example
Solve x² .
- Factor: .
- Set each factor equal to zero.
Answer: or
A common mistake to avoid
Stopping after factoring instead of setting each factor equal to zero.
Your turn: try a free question
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Curriculum and sources
The key ideas and worked example come from PrepSAT's reviewed lesson curriculum. These College Board resources define the SAT scope; PrepSAT is an independent practice resource and is not affiliated with College Board.