A quadratic equation is the first big step past straight-line algebra. It contains a squared unknown, it can have two answers, and its graph bends into a curve. Students meet it in Secondary 4 math, and it shows up again in physics, economics, and every calculus course after that.
This guide answers the question “what is a quadratic equation” from the ground up. It covers the standard form, the discriminant, three algebraic methods to solve quadratic equations, and worked examples you can check yourself.
Quadratic Equation Definition: The Standard Form ax² + bx + c = 0
A quadratic equation is a polynomial equation in which the highest power of the variable is 2. If you need a refresher on what an equation is in general, start there; this page focuses on the degree two case.
Every quadratic can be rearranged into the standard form of a quadratic equation:
ax² + bx + c = 0
Mathematicians also call it an equation of degree 2 or a second-degree equation. The word “quadratic” comes from the Latin quadratus, meaning square, because the unknown is squared.
Inside the Quadratic Expression: Coefficients, Constant Term and Variable
The left side of the standard form, ax² + bx + c, is a quadratic expression (also called a quadratic polynomial). Setting that expression equal to zero turns it into an equation you can solve.
Each part has a name:
- ax² is the quadratic term, and a is its coefficient
- bx is the linear term, with b as its coefficient
- c is the constant term, a number with no variable attached
- x is the variable, the unknown value you solve for
The coefficients a, b, and c are real numbers. They can be positive, negative, or fractions; only a has a restriction.
Why a ≠ 0: A Second-Degree Equation, Not a Linear Equation
The coefficient a can never equal zero. If a = 0, the x² term disappears and 0x² + bx + c = 0 collapses into bx + c = 0, which is a linear equation. Linear equations have one solution, and you handle them with the inverse operations covered in our guide to solving simple equations.
The b and c terms, by contrast, can be zero. 3x² − 12 = 0 has no linear term and x² − 7x = 0 has no constant term, yet both remain quadratic because the second degree term is still there.
Quadratics also have three common written forms:
| Form | Equation | What it shows |
|---|---|---|
| Standard form | ax² + bx + c = 0 | The coefficients a, b, c |
| Factored form (intercept form) | a(x − r₁)(x − r₂) = 0 | The roots r₁ and r₂ |
| Vertex form | a(x − h)² + k = 0 | The vertex (h, k) of the parabola |
Roots of a Quadratic Equation: What the Discriminant Tells You
The roots of a quadratic equation are the values of x that make it true. They are also called solutions or zeros. A quadratic has at most two roots, and one number predicts what kind you will get before you solve anything: the discriminant.
The discriminant is Δ = b² − 4ac, the expression under the square root in the quadratic formula. Its sign decides the nature of roots.
Two Real Roots, a Double Root or Complex Roots
The discriminant sorts every quadratic equation into one of three cases:
- Positive discriminant: two distinct real roots. For x² − 5x + 6 = 0, Δ = 25 − 24 = 1, so there are two real solutions (x = 2 and x = 3).
- Zero discriminant: one double root. For x² − 4x + 4 = 0, Δ = 16 − 16 = 0, giving the repeated root x = 2.
- Negative discriminant: two complex roots. For x² + 2x + 5 = 0, Δ = 4 − 20 = −16. No real number squares to −16, so the complex solutions, written with complex numbers, are x = −1 ± 2i, where i is the imaginary number √−1.
When the coefficients are rational, a positive discriminant that is a perfect square gives rational roots; any other positive value gives irrational roots. Complex roots of an equation with real coefficients always come in conjugate pairs, like −1 + 2i and −1 − 2i. Two different quadratics can also share common roots: x² − 5x + 6 and x² − 4x + 4 both have x = 2.
Sum and Product of Roots with Vieta’s Formulas
Vieta’s formulas link the roots straight back to the coefficients. For ax² + bx + c = 0 with roots r₁ and r₂, the sum of the roots is r₁ + r₂ = −b/a and their product is r₁ × r₂ = c/a.
Take x² − 5x + 6 = 0 again. The sum is 5, and the product is 6, and indeed 2 + 3 = 5 and 2 × 3 = 6. This sum and product of roots relationship is a fast way to check an answer, and it lets you build a quadratic from two known roots.
The Graph of a Quadratic Function: Where the Parabola Crosses the x-Axis
The quadratic function y = ax² + bx + c draws a U-shaped curve called a parabola. Solving the equation ax² + bx + c = 0 means asking where that graph meets the x-axis, so the roots are the x-intercepts.
The discriminant shows up on the graph too. Two x-intercepts mean a positive discriminant. A parabola that just touches the axis at its vertex, on the axis of symmetry, means Δ = 0. A parabola that never reaches the axis means a negative discriminant. The sign of a sets the direction: a positive a opens upward and has a minimum value, while a negative a opens downward and has a maximum value.
Graphing is a good way to estimate solutions, but read the intercepts carefully; a sketch rarely gives exact answers.
How to Solve Quadratic Equations: 3 Algebraic Methods
There are three algebraic ways to solve quadratic equations. Factoring is fastest when it works, completing the square works every time, and the quadratic formula is completing the square done once and for all. All three give the same roots.
Factoring and the Zero Product Property
Factoring rewrites the quadratic expression as a product of two binomials. Once the equation reads (something) × (something) = 0, the zero product property says at least one factor must be equal to zero.
Example: x² − 5x + 6 = 0. Look for two numbers whose product is 6 and whose sum is −5: −2 and −3. So x² − 5x + 6 = (x − 2)(x − 3) = 0, and the roots are x = 2 and x = 3.
Factorization depends on spotting factors quickly. It is the method to try first, but many equations, like x² + x − 1 = 0, have no tidy factors.
Completing the Square, Step by Step
Completing the square turns the left side into a perfect square, then takes a square root of both sides. It goes back to al-Khwarizmi, the 9th-century Baghdad scholar whose book gave algebra its name.
Example: x² + 6x + 5 = 0.
- Move the constant: x² + 6x = −5
- Add (6 ÷ 2)² = 9 to both sides: x² + 6x + 9 = 4
- Write the perfect square: (x + 3)² = 4
- Take the square root: x + 3 = ±2
- Solve: x = −1 or x = −5
Substitute both values back into the original equation to check them: (−1)² + 6(−1) + 5 = 0 and (−5)² + 6(−5) + 5 = 0.
Using the Quadratic Formula
The quadratic formula solves any quadratic equation in standard form:
x = (−b ± √(b² − 4ac)) / 2a
The ± sign (read “plus or minus”) produces both roots at once. The Indian mathematician Brahmagupta described a rule for this in 628, and the modern formula comes from completing the square on ax² + bx + c = 0.
Two habits prevent most errors: rearrange the equation into standard form first, and calculate the discriminant before anything else. If Δ is negative, you already know there are no real solutions.
For the arithmetic under the radical, see our guide to calculating a square root.
Practise the Quadratic Formula with a Math Tutor
A Tutorax tutor can start online within 24 to 48 hours. We match your child with a math tutor who works through the quadratic formula at their pace.
Quadratic Equation Examples: Real-Life Problems and a Worked Solution
Quadratic equations model anything that rises and falls, or any area built from two related lengths. These quadratic equation examples show how a word problem turns into ax² + bx + c = 0.
An area problem: a garden’s length is 3 m more than its width, and its area is 40 m². With width x, x(x + 3) = 40 becomes x² + 3x − 40 = 0, so (x + 8)(x − 5) = 0. A negative width is impossible, so the garden is 5 m by 8 m.
A profit problem: if a business’s profit is P = −2x² + 40x − 150, the break-even points solve −2x² + 40x − 150 = 0, or x² − 20x + 75 = 0. The roots, x = 5 and x = 15, mark the range where profit is positive.
Students who fell behind on these word problems during the year can catch up in math summer school.
Projectile Motion: Finding When a Ball Lands
Projectile motion is the classic physics application. A ball thrown straight up from the ground at 9.8 m/s has a height of h = −4.9t² + 9.8t metres after t seconds.
The ball lands when its height returns to zero: −4.9t² + 9.8t = 0. Factoring gives t(−4.9t + 9.8) = 0, so t = 0 (the throw) or t = 2 seconds (the landing). The vertex at t = 1 gives the maximum height, 4.9 m.
The same equation, with air resistance ignored, describes a basketball shot or a soccer kick.
A physics tutor can show your teen where these real-life applications appear on their tests.
A Worked Example from Secondary 4 Math
Secondary 4 is the year second-degree functions become a core part of the math program. Here is a worked example at that level: solve 2x² − 4x − 6 = 0.
- Identify a = 2, b = −4, c = −6
- Calculate the discriminant: Δ = (−4)² − 4(2)(−6) = 16 + 48 = 64
- Since Δ is a positive perfect square, expect two rational roots
- Apply the formula: x = (4 ± √64) / 4 = (4 ± 8) / 4
- Solve: x = 3 or x = −1, then check both in the original equation
The common mistakes at this level are predictable: dropping the negative sign on b, dividing only part of the numerator by 2a, and skipping the check. Practising these steps before a test is the heart of good exam preparation.
Get Secondary 4 Math Help, at Home or Online
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Most students who struggle with quadratics are missing one earlier skill, usually factoring or negative numbers, not the whole chapter.
Need Help Solving Quadratic Equations? Work with a Math Tutor
Solving quadratic equations builds on fractions, negatives, factoring, and square roots all at once. A gap in any one of them shows up here first. A math tutor finds that gap and rebuilds it before the next chapter.
From Stuck to Confident: Checking Solutions with a Tutor
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Tutors teach students to check every solution, so a correct answer becomes a habit rather than a guess. Grades are the proof; confidence is what changes first.
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