ACT Math: Quadratics
Quadratics is a heavily tested Algebra and Functions topic on the ACT, connecting algebraic solving methods to the geometry of a parabola's graph. This free, complete lesson covers solving quadratic equations by factoring, the quadratic formula and the discriminant, vertex form and graphing parabolas, and building a quadratic from its roots while connecting an equation to its factors and graph. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback, in the same four-answer-choice, calculator-allowed format as the current ACT. Everything here is free, and you can keep practicing afterward with the full ACT Math Question Bank linked below.
1. Solving Quadratics by Factoring
If a quadratic factors nicely, factoring is the fastest way to solve it, using the zero product property: if two factors multiply to zero, at least one of them must be zero.
Worked Example: Solve x² − 7x + 12 = 0 by factoring.
Find two numbers that multiply to 12 and add to −7: −3 and −4.
(x − 3)(x − 4) = 0
x = 3 or x = 4
1. Solve by factoring: x² − 9x + 20 = 0
Explanation: Find two numbers that multiply to 20 and add to −9: −4 and −5. (x − 4)(x − 5) = 0, so x = 4 or x = 5.
2. Solve by factoring: x² + 3x − 10 = 0
Explanation: Find two numbers that multiply to −10 and add to 3: 5 and −2. (x + 5)(x − 2) = 0, so x = −5 or x = 2.
3. Solve by factoring: x² − 16 = 0
Explanation: This is a difference of squares: x² − 16 = (x − 4)(x + 4) = 0, so x = 4 or x = −4.
4. Solve by factoring: 2x² − 5x − 3 = 0
Explanation: Multiply a and c: 2(−3) = −6. Find two numbers that multiply to −6 and add to −5: −6 and 1. Rewrite and group: 2x² − 6x + x − 3 = 2x(x − 3) + 1(x − 3) = (2x + 1)(x − 3) = 0, so x = −12 or x = 3.
5. Solve by factoring: x² − 8x = 0
Explanation: Factor out x: x(x − 8) = 0, so x = 0 or x = 8. Don't divide both sides by x, doing so loses the solution x = 0.
2. The Quadratic Formula and the Discriminant
The quadratic formula solves any equation of the form ax² + bx + c = 0:
x = −b ± √(b² − 4ac)2a
The expression under the square root, b² − 4ac, is the discriminant. It reveals the number of real solutions without needing to fully solve the equation:
| Discriminant | Number of real solutions |
|---|---|
| Positive | 2 distinct real solutions |
| Zero | 1 repeated real solution |
| Negative | 0 real solutions |
Worked Example: Solve 2x² + 3x − 5 = 0 using the quadratic formula.
a = 2, b = 3, c = −5. Discriminant: 3² − 4(2)(−5) = 9 + 40 = 49.
x = −3 ± √494 = −3 ± 74
x = 1 or x = −2.5
1. Use the quadratic formula to solve x² − 2x − 15 = 0.
Explanation: a = 1, b = −2, c = −15. Discriminant: 4 + 60 = 64. x = 2 ± 82, giving x = 5 or x = −3.
2. What is the discriminant of 3x² − 4x + 5 = 0?
Explanation: b² − 4ac = (−4)² − 4(3)(5) = 16 − 60 = −44.
3. A quadratic equation has a discriminant of −44. How many real solutions does it have?
Explanation: A negative discriminant means the square root of a negative number would be required, so there are no real solutions.
4. A quadratic equation has a discriminant of 0. How many real solutions does it have?
Explanation: When the discriminant is 0, the ± in the quadratic formula produces the same value both times, giving exactly one repeated real solution.
5. Use the quadratic formula to solve x² + 4x + 1 = 0. Express the solutions in simplest radical form.
Explanation: a = 1, b = 4, c = 1. Discriminant: 16 − 4 = 12. x = −4 ± √122 = −4 ± 2√32 = −2 ± √3.
3. Vertex Form and Graphing Parabolas
Every quadratic graphs as a parabola. Writing it in vertex form, y = a(x − h)² + k, makes the vertex (h, k) immediately visible. If a > 0, the parabola opens upward and the vertex is a minimum; if a < 0, it opens downward and the vertex is a maximum. From standard form y = ax² + bx + c, the vertex's x-coordinate is always x = −b2a, also called the axis of symmetry.
a > 0: opens upward, vertex is a minimum
Worked Example: Identify the vertex of y = 2(x − 3)² + 7, and state whether it's a maximum or minimum.
The vertex is (3, 7). Since a = 2 > 0, the parabola opens upward, so this vertex is a minimum.
1. What is the vertex of y = −3(x + 4)² + 7?
Explanation: Rewrite x + 4 as x − (−4), so h = −4 and k = 7, giving vertex (−4, 7).
2. Is the vertex of y = −3(x + 4)² + 7 a maximum or minimum?
Explanation: Since a = −3 < 0, the parabola opens downward, making the vertex a maximum.
3. Find the vertex of y = x² − 6x + 5.
Explanation: x = −(−6)2(1) = 3. Then y = 3² − 6(3) + 5 = 9 − 18 + 5 = −4. Vertex: (3, −4).
4. What is the axis of symmetry of y = 2x² − 8x + 3?
Explanation: x = −(−8)2(2) = 84 = 2.
5. A parabola has vertex (2, −5) and opens upward. What is the minimum value of the function?
Explanation: The minimum value of a parabola that opens upward is the y-coordinate of its vertex, −5.
4. Building Quadratics from Roots
If a quadratic has roots (zeros) r1 and r2, it can be written in factored form as y = a(x − r1)(x − r2) for any nonzero a. This connects an equation directly to its graph: the roots are exactly where the parabola crosses the x-axis, and the axis of symmetry always sits exactly halfway between them.
Worked Example: Write a quadratic equation in factored form with roots x = 3 and x = −1 (using a = 1).
y = (x − 3)(x − (−1)) = (x − 3)(x + 1)
1. Write a quadratic equation in standard form with roots x = 5 and x = −2 (using a = 1).
Explanation: y = (x − 5)(x + 2) = x² + 2x − 5x − 10 = x² − 3x − 10.
2. A quadratic function has zeros at x = −4 and x = 6. What is the x-coordinate of the vertex (the axis of symmetry)?
Explanation: The axis of symmetry lies exactly halfway between the two roots: −4 + 62 = 22 = 1.
3. The graph of y = (x − 2)(x + 5) crosses the x-axis at which points?
Explanation: Setting each factor to 0: x − 2 = 0 gives x = 2, and x + 5 = 0 gives x = −5.
4. A ball's height is modeled by h(t) = −16t² + 64t, where t is time in seconds. At what time does the ball land (h = 0, for t > 0)?
Explanation: −16t² + 64t = 0. Factor: −16t(t − 4) = 0, so t = 0 or t = 4. Since t = 0 is the starting moment, the ball lands at t = 4 seconds.
Common Mistakes to Avoid
- Dividing both sides by x instead of factoring it out. In an equation like x² − 8x = 0, dividing by x loses the solution x = 0. Always factor instead.
- Forgetting the ± when using the quadratic formula. A quadratic formula problem almost always produces two solutions, one from the + and one from the −, missing the second solution is a very common error.
- Mixing up the sign inside vertex form. In y = a(x − h)² + k, an expression like (x + 4)² means h = −4, not 4, since it's really (x − (−4))².
- Confusing what a>0 versus a<0 means for a parabola's vertex. A positive leading coefficient opens the parabola upward with a minimum vertex; a negative leading coefficient opens it downward with a maximum vertex.
- Assuming the discriminant's sign tells you which solutions exist, without checking the actual value. The discriminant only tells you how many real solutions exist and whether they're distinct, you still need the full quadratic formula to find their actual values.
Frequently Asked Questions
Which method should I use to solve a quadratic: factoring or the quadratic formula?
Try factoring first, since it's usually the fastest when the numbers work out cleanly. If you can't spot integer factors quickly, or you're unsure, the quadratic formula always works and is reliable under time pressure.
Why is the discriminant useful if I still need to fully solve the equation for the actual values?
The discriminant answers "how many real solutions" instantly, without doing the rest of the quadratic formula. This is especially useful for conceptual questions that only ask about the number or nature of solutions, not their exact values, saving significant time.
How are a quadratic's roots, its factored form, and its graph all connected?
They describe the same information three different ways: the roots are the x-values where y = 0, the factored form y = a(x − r₁)(x − r₂) directly encodes those roots, and the graph crosses the x-axis exactly at those same points. Recognizing this connection often lets you skip algebra entirely on graph-based questions.
Where can I practice more problems like these?
The Quadratics quizzes in the ACT Math Question Bank include additional original problems on this topic, along with quizzes covering every other topic tested on the ACT Math section.