If linear equations are the “walk” of algebra, quadratic equations are the first real “run.” They show up the moment a real situation involves something multiplied by itself, a square area, a ball thrown into the air, a profit curve that rises and then falls. A quadratic equation is simply an equation where the highest power of the unknown is 2, and that one extra power changes everything: instead of a single answer, you can get two answers, one answer, or none at all.
This guide builds on our companion article on algebraic equations and goes deep into this one specific type. You’ll get every solving method side by side, a full explanation of the discriminant, more than twenty worked examples ranging from simple to competition-level, and the mistakes that quietly cost students the most marks.
What Is a Quadratic Equation?
Featured snippet answer: A quadratic equation is a polynomial equation of degree 2, meaning the highest power of the variable is 2. Its standard form is ax² + bx + c = 0, where a, b, and c are constants and a is not zero. It can have zero, one, or two real solutions. Encyclopedia Britannica’s entry on quadratic equations and Wikipedia’s quadratic equation article both give a concise historical and mathematical overview if you want additional reference points.
Standard form: ax² + bx + c = 0, where a ≠ 0
If a were allowed to be zero, the x² term would vanish and the equation would collapse into a linear equation, which is why that condition is always stated.
Why the graph is a parabola
Picture plotting y = x² for a few values of x: at x = -2, y = 4; at x = -1, y = 1; at x = 0, y = 0; at x = 1, y = 1; at x = 2, y = 4. The output shrinks toward zero and grows again symmetrically on both sides, producing a U-shaped curve called a parabola. Every quadratic equation’s graph is a parabola, just shifted, stretched, or flipped depending on a, b, and c. Wolfram MathWorld’s parabola entry and Wikipedia’s parabola article both cover the geometric definition in more depth, including the focus and directrix, if you want to go beyond the algebra.
Parts of a Quadratic Equation
Take 3x² – 5x + 2 = 0:
- Leading coefficient (a): 3, the number multiplying x²
- Linear coefficient (b): -5, the number multiplying x
- Constant term (c): 2
- Roots/solutions: the values of x that make the equation true
For a well-documented history of how this equation type was solved centuries before symbolic algebra existed, the Smithsonian’s National Museum of American History and Wikipedia’s history of algebra article both trace the Babylonian and Islamic Golden Age contributions that led to the modern formula.
The Discriminant: Predicting the Roots Before Solving
The discriminant is the expression b² – 4ac, taken from inside the square root of the quadratic formula. It tells you what kind of roots to expect before you do any further work.
| Discriminant Value | Nature of Roots | Graph Behavior |
|---|---|---|
| b² – 4ac > 0 | Two distinct real roots | Parabola crosses the x-axis twice |
| b² – 4ac = 0 | One repeated real root | Parabola touches the x-axis at exactly one point |
| b² – 4ac < 0 | Two complex (non-real) roots | Parabola never touches the x-axis |
Worked example: For x² – 4x + 4 = 0, the discriminant is (-4)² – 4(1)(4) = 16 – 16 = 0, so there is exactly one repeated root. Factoring confirms this: (x-2)² = 0, so x = 2.
See Wikipedia’s entry on the discriminant and the NIST Digital Library of Mathematical Functions for formal treatments of this same result.
Methods for Solving Quadratic Equations
Paul’s Online Math Notes from Lamar University is a widely used reference that walks through each of these methods step by step if you want a second worked explanation alongside this one.
| Method | Best For | Advantage | Limitation |
|---|---|---|---|
| Factoring | Equations that split into neat integer factors | Fast, shows roots directly | Doesn’t work when roots aren’t rational |
| Completing the Square | Any quadratic, especially for deriving vertex form | Always works, reveals the vertex | More steps than factoring |
| Quadratic Formula | Any quadratic without exception | Universal, guaranteed to work | Requires careful arithmetic with the square root |
| Graphical Method | Understanding and estimating roots visually | Builds intuition, good for checking | Less precise without graphing tools |
The Quadratic Formula
x = (-b ± √(b² – 4ac)) / 2a
This single formula solves every quadratic equation ever written, whether or not it factors neatly. Wolfram MathWorld’s derivation and Wikipedia’s quadratic formula article both show exactly how completing the square on the general form produces this result, and the NIST Digital Library of Mathematical Functions gives the formal reference definition used across scientific computing.
Factoring Method (worked example)
Wikipedia’s article on factorization covers the general theory behind this technique if you want the broader mathematical context.
Solve x² – x – 6 = 0 Look for two numbers that multiply to -6 and add to -1: those are -3 and 2. (x – 3)(x + 2) = 0 x = 3 or x = -2
Completing the Square (worked example)
For a deeper look at why this method works geometrically (literally completing a square shape), see Wikipedia’s article on completing the square, which includes the historical geometric construction.
Solve x² + 8x + 7 = 0 Move the constant: x² + 8x = -7 Take half of 8, square it (16), and add to both sides: x² + 8x + 16 = -7 + 16 (x + 4)² = 9 x + 4 = ±3 x = -1 or x = -7
Graphical Method (concept)
Plot y = ax² + bx + c and read off where the curve crosses the x-axis. This won’t give exact fractional roots by hand, but it’s the fastest way to see how many real roots exist and roughly where they sit, which is often exactly what a word problem needs. Desmos’s free online graphing calculator is a reliable tool for plotting any quadratic instantly and checking your algebraic answer against the graph.
Vertex Form and the Vertex
Every quadratic can be rewritten as y = a(x – h)² + k, where (h, k) is the vertex, the turning point of the parabola. The x-coordinate of the vertex is always h = -b / (2a). Wikipedia’s article on the vertex of a parabola covers this concept in the broader context of conic sections.
Worked example: For y = x² – 6x + 5, h = -(-6)/(2×1) = 3. Substituting back, k = 3² – 6(3) + 5 = 9 – 18 + 5 = -4. Vertex: (3, -4).
20 Step-by-Step Solved Examples
Easy
- x² – 9 = 0 → x = 3 or x = -3
- x² – 16 = 0 → x = 4 or x = -4
- x² + 5x = 0 → x(x+5) = 0 → x = 0 or x = -5
- x² – 3x = 0 → x(x-3) = 0 → x = 0 or x = 3
- x² – 6x + 9 = 0 → (x-3)² = 0 → x = 3
Medium
- x² – 7x + 10 = 0 → (x-2)(x-5) = 0 → x = 2, 5
- x² + 3x – 10 = 0 → (x+5)(x-2) = 0 → x = -5, 2
- 2x² – 5x – 3 = 0 → (2x+1)(x-3) = 0 → x = -1/2, 3
- x² – 2x – 8 = 0 → (x-4)(x+2) = 0 → x = 4, -2
- 4x² – 9 = 0 → (2x-3)(2x+3) = 0 → x = 3/2, -3/2
Using the Quadratic Formula
- x² + 3x + 1 = 0 → x = (-3 ± √(9-4)) / 2 = (-3 ± √5) / 2
- 2x² – 4x – 3 = 0 → x = (4 ± √(16+24)) / 4 = (4 ± √40) / 4 = (2 ± √10) / 2
- x² + x – 1 = 0 → x = (-1 ± √5) / 2 (this is related to the golden ratio)
Word Problems
- Area problem: A rectangular garden is 4 m longer than it is wide, and its area is 96 m². Find its dimensions. Let width = x, length = x + 4. x(x+4) = 96 → x² + 4x – 96 = 0 → (x+12)(x-8) = 0 → x = 8 Width = 8 m, length = 12 m.
- Projectile problem: A ball is thrown upward, and its height in meters is given by h = -5t² + 20t, where t is time in seconds. When does it hit the ground? Set h = 0: -5t² + 20t = 0 → -5t(t – 4) = 0 → t = 0 or t = 4 The ball hits the ground at t = 4 seconds.
- Number problem: The product of two consecutive positive integers is 132. Find the integers. Let the integers be x and x+1. x(x+1) = 132 → x² + x – 132 = 0 → (x+12)(x-11) = 0 → x = 11 The integers are 11 and 12.
- Frame problem: A photo is 8 cm by 10 cm, and a frame of uniform width around it increases the total area to 143 cm². Find the width of the frame. Let width = x. Total dimensions: (8+2x) by (10+2x). (8+2x)(10+2x) = 143 → 80 + 36x + 4x² = 143 → 4x² + 36x – 63 = 0 Using the quadratic formula: x = (-36 ± √(1296+1008)) / 8 = (-36 ± √2304) / 8 = (-36 ± 48) / 8 x = 1.5 (taking the positive root) The frame is 1.5 cm wide.
- Speed/time problem: A car covers 180 km. If its speed had been 5 km/h more, the journey would have taken 1 hour less. Find the original speed. Let speed = x. 180/x – 180/(x+5) = 1 180(x+5) – 180x = x(x+5) → 900 = x² + 5x → x² + 5x – 900 = 0 x = (-5 ± √(25+3600)) / 2 = (-5 ± 60.2) / 2 ≈ 27.6 km/h (rounded)
Competition Level
- If one root of x² – 7x + k = 0 is 3, find k and the other root. Substitute x=3: 9 – 21 + k = 0 → k = 12 Equation becomes x² – 7x + 12 = 0 → (x-3)(x-4) = 0 → other root is 4
- For what value of k does x² + kx + 9 = 0 have equal roots? Equal roots means discriminant = 0: k² – 4(1)(9) = 0 → k² = 36 → k = ±6
Real-Life Applications of Quadratic Equations
- Physics: projectile motion, where height over time follows a quadratic curve, the same relationship described in NASA’s Glenn Research Center notes on projectile motion and Wikipedia’s article on projectile motion
- Business: profit functions, where revenue minus cost often produces a parabola with a maximum profit point, a concept covered in Investopedia’s explanation of profit maximization
- Architecture and design: parabolic arches and satellite dish shapes are literally quadratic curves in physical form, as described in Britannica’s entry on the parabola and documented by the U.S. National Park Service in its notes on the Gateway Arch’s catenary-parabola design
- Engineering: optimizing area or volume given a fixed amount of material
- Sports: modeling the trajectory of a thrown or kicked ball
Common Mistakes with Quadratic Equations
- Forgetting the ± sign in the quadratic formula, which loses one of the two roots.
- Applying the quadratic formula with the wrong signs for b when b is already negative.
- Assuming every quadratic factors over the integers, when many require the formula instead.
- Dividing an equation by x, accidentally losing the root x = 0.
- Forgetting to check the discriminant before assuming real roots exist.
- Making an arithmetic slip inside the square root, especially with negative numbers.
- Mixing up the vertex formula and applying -b/(2a) as though it were a root instead of the axis of symmetry.
- Not simplifying the final answer, leaving an unreduced fraction or square root.
Frequently Asked Questions
1. What is a quadratic equation? A polynomial equation of degree 2, written in standard form as ax² + bx + c = 0, where a ≠ 0.
2. How many solutions does a quadratic equation have? Up to two real solutions, exactly one repeated real solution, or two complex solutions, depending on the discriminant.
3. What is the quadratic formula? x = (-b ± √(b² – 4ac)) / 2a, which solves any quadratic equation regardless of whether it factors. Wolfram MathWorld has the full derivation.
4. What is the discriminant? The expression b² – 4ac, which predicts whether the roots are two distinct real numbers, one repeated real number, or two complex numbers. See the NIST Digital Library of Mathematical Functions for the formal definition.
5. Can a quadratic equation have no real solutions? Yes, when the discriminant is negative, the equation has two complex roots and the parabola never touches the x-axis. See Wikipedia’s article on complex numbers for how these roots are represented.
6. What is the difference between a root and a zero? They mean the same thing in this context, a value of x that makes the equation equal to zero.
7. What is the vertex of a parabola? The highest or lowest point on the curve, found at x = -b / (2a). Purplemath’s guide to vertex form and Wikipedia’s vertex (geometry) article both cover several additional worked conversions.
8. How do you know if a quadratic factors nicely? If the discriminant is a perfect square, the equation factors into rational numbers; otherwise, use the quadratic formula.
9. What is completing the square used for? It converts a quadratic into vertex form and derives the quadratic formula itself; it is also useful for solving equations directly.
10. Why is the coefficient a never allowed to be zero? Because if a = 0, the x² term disappears and the equation becomes linear, not quadratic.
Glossary
- Discriminant: the value b² – 4ac, used to determine the nature of the roots.
- Parabola: the U-shaped curve that is the graph of any quadratic equation.
- Root: a value of the variable that satisfies the equation.
- Vertex: the turning point of a parabola.
- Vertex form: the form y = a(x-h)² + k, showing the vertex directly.
Practice Questions
- Solve: x² – 5x + 6 = 0
- Solve: x² + 2x – 15 = 0
- Solve using the quadratic formula: 2x² + 3x – 2 = 0
- Find the discriminant of x² – 4x + 5 = 0 and describe the nature of its roots
- A rectangle has area 60 and length 4 more than its width. Find the width.
Answer key: 1) x = 2, 3 2) x = -5, 3 3) x = 1/2, -2 4) Discriminant = -4, two complex roots 5) width = 6
Summary
- A quadratic equation has the form ax² + bx + c = 0, with a ≠ 0.
- The discriminant b² – 4ac predicts the number and type of roots before you solve.
- Factoring is fastest when it works; completing the square always works and reveals the vertex; the quadratic formula always works without exception.
- Real-life uses range from projectile motion to profit optimization to architectural curves.
Conclusion
Quadratic equations reward the same discipline as linear ones, just with an extra layer: check the discriminant, pick the right method, and always double back to verify both roots. Once factoring, completing the square, and the quadratic formula all feel like different paths to the same destination rather than three separate topics to memorize, quadratics stop being intimidating and start being one of the most useful tools in the entire algebra toolkit.