PSAT/NMSQT Math: Quadratic Equations
Solving a quadratic equation means finding the value or values of x that make it true, and the PSAT/NMSQT expects you to be comfortable with several methods. This free, complete lesson covers solving by factoring, rearranging an equation into standard form first, completing the square, and constructing a quadratic equation directly from its roots using Vieta's formulas. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback and full explanations. Everything here is free, and you can keep practicing afterward with the linked quiz below.
1. Solving by Factoring
If a quadratic can be factored into two binomials, set each factor equal to zero and solve, using the Zero Product Property: if the product of two factors is 0, at least one of the factors must itself be 0.
Worked Example: Solve x² + 10x + 16 = 0 by factoring.
Find two numbers that multiply to 16 and add to 10: 2 and 8.
(x + 2)(x + 8) = 0, so x = −2 or x = −8
1. Solve x² − 5x + 6 = 0 by factoring.
Explanation: Find two numbers that multiply to 6 and add to −5: −2 and −3. (x − 2)(x − 3) = 0, so x = 2 or x = 3.
2. Identify the correct factorization of x² + 10x + 16 = 0.
Explanation: The numbers 2 and 8 multiply to 16 and add to 10, so the correct factorization is (x + 2)(x + 8).
3. Solve x² + 4x = 5 (hint: rearrange into standard form first).
Explanation: Rearranged to standard form: x² + 4x − 5 = 0. Factoring: (x + 5)(x − 1) = 0, so x = −5 or x = 1.
2. Rearranging to Standard Form and Factoring Out the GCF
Before you can factor or apply the quadratic formula, an equation must be in standard form, ax² + bx + c = 0. Move every term to one side first. If every term shares a common factor, factor out the GCF first, this often reveals x = 0 as one solution.
Worked Example: Solve 3x² − 12x = 0.
Factor out the GCF, 3x:
3x(x − 4) = 0, so x = 0 or x = 4
1. Solve 3x² − 12x = 0 by factoring out the GCF. (worked example above)
Explanation: 3x(x − 4) = 0 gives x = 0 or x = 4 by the Zero Product Property.
2. Solve 5x² + 15x = 0.
Explanation: Factor out 5x: 5x(x + 3) = 0, so x = 0 or x = −3.
3. Rearrange 2x² = 8x − 6 into standard form (ax² + bx + c = 0).
Explanation: Moving every term to the left side: 2x² − 8x + 6 = 0.
3. Completing the Square
Completing the square converts a quadratic from standard form into (x − h)² = k form, which can then be solved by taking a square root. Take half of the x-coefficient, square it, and add that value to both sides.
Worked Example: Convert x² − 6x + 8 = 0 into (x − h)² = k form.
Move the constant: x² − 6x = −8. Half of −6 is −3, and (−3)² = 9. Add 9 to both sides:
x² − 6x + 9 = 1 → (x − 3)² = 1
1. Using (x − 3)² = 1 from the worked example, solve for x.
Explanation: Taking the square root of both sides: x − 3 = ±1, so x = 3 + 1 = 4 or x = 3 − 1 = 2.
2. What value must be added to both sides to complete the square for x² + 8x = 3?
Explanation: Half of 8 is 4, and 4² = 16, so 16 must be added to both sides.
3. Complete the square for x² + 8x = 3, then solve for x.
Explanation: x² + 8x + 16 = 19 → (x + 4)² = 19 → x + 4 = ±√19 → x = −4 ± √19.
4. Constructing Equations from Roots and Vieta's Formulas
If a quadratic's roots are r₁ and r₂, the equation can be written directly as (x − r₁)(x − r₂) = 0. Vieta's formulas give a shortcut: for x² + bx + c = 0, the sum of the roots equals −b, and the product of the roots equals c.
Worked Example: Write a quadratic equation with roots x = −2 and x = 5.
(x − (−2))(x − 5) = 0 → (x + 2)(x − 5) = 0 → x² − 3x − 10 = 0
1. Write a quadratic equation with roots x = 2 and x = −3/2, in the form (x − r₁)(x − r₂) = 0 expanded.
Explanation: (x − 2)(x + 3/2) = 0. Multiplying through by 2 to clear the fraction: (x − 2)(2x + 3) = 0 → 2x² − x − 6 = 0.
2. Using Vieta's formulas, find the sum of the roots of x² − 8x + 12 = 0.
Explanation: For x² + bx + c = 0, the sum of the roots is −b. Here b = −8, so the sum is −(−8) = 8.
3. Using Vieta's formulas, find the product of the roots of x² − 8x + 12 = 0.
Explanation: For x² + bx + c = 0, the product of the roots is c. Here c = 12, so the product of the roots is 12.
Common Mistakes to Avoid
- Trying to factor before rearranging into standard form. Every term must be on one side, set equal to 0, before factoring or applying the quadratic formula.
- Forgetting the x = 0 solution when factoring out a GCF that includes a variable. Factoring 3x(x − 4) = 0 gives two solutions, x = 0 and x = 4, not just one.
- Adding the "complete the square" value to only one side of the equation. Whatever you add must be added to both sides to keep the equation balanced.
- Forgetting the ± when taking a square root. (x − h)² = k has two solutions, x = h + √k and x = h − √k, unless k = 0.
- Mixing up which Vieta's formula gives the sum versus the product. For x² + bx + c = 0, sum = −b, and product = c, a sign flip only applies to the sum.
Frequently Asked Questions
How do I decide which method to use, factoring or completing the square?
Try factoring first, since it's usually faster when the quadratic has nice integer roots. If the numbers don't factor cleanly, completing the square (or the quadratic formula) will always work.
What are Vieta's formulas actually useful for on the PSAT?
They let you find the sum or product of a quadratic's roots directly from its coefficients, without ever solving the equation, which is especially fast for multiple-choice questions asking only for the sum or product.
Where can I practice more problems like these?
The Quadratic Equations quiz in the PSAT/NMSQT Math Question Bank includes additional original problems on this topic, along with quizzes covering every other Heart of Algebra, Advanced Math, Problem Solving and Data Analysis, and Geometry and Trigonometry skill tested on the PSAT/NMSQT. You can also browse the full PSAT/NMSQT Math Question Bank.