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Quadratic Equations for the PSAT/NMSQT: Complete Study Guide + Free Practice Problems
Everything the PSAT/NMSQT tests about quadratics in one place: solving by factoring, the quadratic formula, completing the square, and graphing parabolas — vertex, axis of symmetry, and intercepts — with step-by-step examples, worked problems, and 3 free practice quizzes.
Practice these quizzes
Free · No signupWhat is a quadratic equation?
A quadratic equation contains a variable raised to the second power, written in standard form as ax² + bx + c = 0. Its graph is a parabola, a U-shaped curve, and it can have zero, one, or two real solutions.
The PSAT tests quadratics two ways: as pure algebra (solve for x) and as a graph (identify the vertex, axis of symmetry, or intercepts). Both skills draw from the same underlying equation.
Solving by factoring
Solve: x² − x − 12 = 0
A quadratic must be set equal to zero before factoring. If the equation is written as x² − x = 12, move the 12 over first — factoring x(x − 1) = 12 and setting each factor equal to 12 is incorrect.
The quadratic formula
Solve: 2x² − 5x − 3 = 0
Completing the square
Rewrite x² + 6x + 4 = 0 by completing the square.
Completing the square is especially useful when the PSAT asks you to rewrite a quadratic in vertex form — the process directly reveals the vertex coordinates.
Vertex & axis of symmetry
What is the vertex of f(x) = 3(x − 2)² + 7?
In vertex form, the h-value is subtracted inside the parentheses. If the equation shows (x + 4)², that means h = −4, not 4 — watch this sign flip carefully.
x- & y-intercepts
| Intercept | How to find it |
|---|---|
| y-intercept | Set x = 0 and solve for y (in standard form, this is simply c) |
| x-intercepts | Set y = 0 and solve; easiest to read directly from factored form |
Find the x-intercepts and y-intercept of f(x) = (x − 5)(x + 2).
The axis of symmetry always passes exactly halfway between the two x-intercepts. If you know both x-intercepts, average them to find the x-coordinate of the vertex without any extra algebra.
Common PSAT traps
- Forgetting to set the equation to zero before factoring: always move every term to one side first.
- Sign errors in vertex form: (x − h) means the vertex x-coordinate is +h, and (x + h) means it's −h.
- Mixing up which form reveals which feature: factored form reveals x-intercepts; vertex form reveals the vertex; standard form reveals the y-intercept directly.
- Dropping the ± in the quadratic formula: most quadratics have two solutions — don't stop after finding just one.
Test day strategy for quadratic equations
| Question signal | Fastest approach |
|---|---|
| Quadratic that looks factorable | Try factoring first — faster than the formula when it works |
| Quadratic with ugly coefficients | Go straight to the quadratic formula |
| Vertex or max/min asked | Look for vertex form first, or complete the square from standard form |
| x-intercepts asked | Look for factored form first, or factor standard form |
| y-intercept asked | Read c directly from standard form, or substitute x = 0 |
Now put it to work
Three quiz sets, each building on the last — start with Quiz 1 and work through in order, or jump straight to the topic you need.
Quadratic Equations
Factoring, the quadratic formula, and completing the square.
Start quiz → Quiz 2Graphing Quadratics 1
Vertex and axis of symmetry.
Start quiz → Quiz 3Graphing Quadratics 2
x- and y-intercepts, and reading parabola graphs.
Start quiz →