Quadratic Equations — Full Course

Methods, step-by-step worked examples, interactive graphs, quizzes, and real-world applications.

What is a quadratic equation?

A quadratic equation is a polynomial of degree 2: \( ax^2 + bx + c = 0 \), where \(a\ne0\). The graph is a parabola; its vertex, axis of symmetry and roots are key features.

Quick formulas

  • Quadratic formula: \( x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
  • Discriminant: \( \Delta = b^2 - 4ac \) — tells nature of roots.
  • Vertex (from standard form): vertex \( (h,k) \) with \( h = -\dfrac{b}{2a},\; k = f(h) \)

Solving Methods (with worked examples)

1. Factoring

Write \(ax^2+bx+c\) as product of two binomials when possible.

Example: Solve \(x^2 - 5x + 6 = 0\). Factor: \((x-2)(x-3)=0\) → solutions: \(x=2,\,x=3\).

2. Completing the square

Rewrite to a perfect square and take square root.

Example: Solve \(x^2 - 6x + 5 = 0\). Move constant: \(x^2-6x=-5\). Add \((6/2)^2=9\) to both sides: \((x-3)^2=4\). So \(x-3=\pm2\) → \(x=5\) or \(x=1\).

3. Quadratic formula (general)

Always works. Plug \(a,b,c\) into the formula.

Example: Solve \(2x^2-4x-6=0\). Use formula: \(x=\dfrac{4\pm\sqrt{(-4)^2-4(2)(-6)}}{4}=\dfrac{4\pm\sqrt{16+48}}{4}=\dfrac{4\pm8}{4}\) → \(x=3\) or \(x=-1\).

The discriminant & complex roots

The discriminant \( \Delta = b^2-4ac \):

  • \( \Delta>0\): two distinct real roots
  • \( \Delta=0\): one repeated real root
  • \( \Delta>0\): two complex conjugate roots
Complex example: Solve \(x^2+2x+5=0\). \(\Delta=4-20=-16\) so \(x=\dfrac{-2\pm\sqrt{-16}}{2}=-1\pm2i\).

Interactive quadratic graph

Enter coefficients \(a,b,c\) and view the parabola and its vertex/roots when real.

Table of common exponents & laws

ExpressionValue / Description
\(2^0\)1
\(2^3\)8
\(10^{-1}\)0.1
\(a^m \cdot a^n\)\(a^{m+n}\)
\((a^m)^n\)\(a^{mn}\)

Practice problems (with solutions)

  1. Problem: Solve \(x^2-4x-5=0\).
    Solution: Factor → \((x-5)(x+1)=0\) ⇒ \(x=5\) or \(x=-1\).
  2. Problem: Solve \(3x^2+6x+2=0\).
    Solution: Use quadratic formula: \(x = \dfrac{-6\pm\sqrt{36-24}}{6} = \dfrac{-6\pm\sqrt{12}}{6} = \dfrac{-6\pm2\sqrt{3}}{6} = -1\pm\dfrac{\sqrt{3}}{3}.\)

Interactive quizzes

Quick quiz

1) What are the roots of \(x^2-4x-5=0\)?



Methods quiz

2) Which method is best for \(x^2-5x+6=0\)?



Trig identities & interactive plot

Compound angle: \(\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B\)

Double angle: \(\sin(2A)=2\sin A\cos A\) and \(\cos(2A)=\cos^2 A-\sin^2 A\)

Animated Visuals: Understanding Quadratic Motion

Visualizing quadratic motion helps students connect equations with real-world behavior. Below is an interactive animation showing projectile motion following a quadratic path.

Adjust velocity and angle, then click Start to see how they affect the parabola.

Explanation

The path follows h(t) = -½gt² + v₀sin(θ)t and x(t) = v₀cos(θ)t, where g = 9.8 m/s². The motion forms a perfect parabola — typical of all projectiles on Earth (ignoring air resistance).

Try It Yourself

  • Change the angle to 30°, 45°, and 60° — observe the horizontal distance (range) and peak height.
  • Notice how increasing velocity stretches the parabola horizontally and vertically.

Interactive Quiz: Interpreting the Animation

1 At what angle does a projectile travel the maximum range in ideal conditions?



2 If velocity doubles, what happens to the range?



Animated Visuals: Understanding Quadratic Motion

Visualizing quadratic motion helps students connect equations with real-world behavior. Below is an interactive animation showing projectile motion following a quadratic path.

Adjust velocity and angle, then click Start to see how they affect the parabola.

Explanation

The path follows h(t) = -½gt² + v₀sin(θ)t and x(t) = v₀cos(θ)t, where g = 9.8 m/s². The motion forms a perfect parabola — typical of all projectiles on Earth (ignoring air resistance).

Try It Yourself

  • Change the angle to 30°, 45°, and 60° — observe the horizontal distance (range) and peak height.
  • Notice how increasing velocity stretches the parabola horizontally and vertically.

Interactive Quiz: Interpreting the Animation

1) At what angle does a projectile travel the maximum range in ideal conditions?



2) If velocity doubles, what happens to the range?



Animated Visuals: Understanding Quadratic Motion

Visualizing quadratic motion helps students connect equations with real-world behavior. Below is an interactive animation showing projectile motion following a quadratic path.

Adjust velocity and angle, then click Start to see how they affect the parabola.

Explanation

The path follows h(t) = -½gt² + v₀sin(θ)t and x(t) = v₀cos(θ)t, where g = 9.8 m/s². The motion forms a perfect parabola — typical of all projectiles on Earth (ignoring air resistance).

Try It Yourself

  • Change the angle to 30°, 45°, and 60° — observe the horizontal distance (range) and peak height.
  • Notice how increasing velocity stretches the parabola horizontally and vertically.

Interactive Quiz: Interpreting the Animation

1) At what angle does a projectile travel the maximum range in ideal conditions?



2) If velocity doubles, what happens to the range?



Quadratic Sequences & Series

Quadratic sequences are number patterns where the differences between consecutive terms do not remain constant, but the second differences are constant. These sequences follow a quadratic pattern described by an equation of the form:

Tn = a n² + b n + c

1. Understanding First and Second Differences

To determine if a sequence is quadratic, calculate the first and second differences:

  • First difference: Subtract each term from the next.
  • Second difference: Subtract each first difference from the next.

If the second differences are equal, the sequence is quadratic.

2. Example: Finding the nth Term

Consider the sequence: 2, 6, 12, 20, 30, ...

  • First differences: 4, 6, 8, 10
  • Second differences: 2, 2, 2 (constant)
  • Since the second difference is 2, we know that 2a = 2 ⟹ a = 1.
  • Assume the nth term has the form Tn = n² + b n + c.
  • For n = 1: 1 + b + c = 2 → b + c = 1
  • For n = 2: 4 + 2b + c = 6 → 2b + c = 2
  • Subtract: b = 1, c = 0

Therefore: Tn = n² + n

3. Applications of Quadratic Sequences

  • Mathematics: Used to model patterns in square numbers, polygonal numbers, and growth trends.
  • Physics: Describes uniformly accelerated motion (distance-time relationships).
  • Architecture: Parabolic designs in bridges and domes often follow quadratic relations.

4. Visualizing Quadratic Sequences

Use the interactive chart below to visualize how coefficients (a, b, c) affect the shape of the sequence curve.

5. Summing Quadratic Sequences (Series)

The sum of the first n terms of a quadratic sequence can be found using:

Sn = a·n(n+1)(2n+1)/6 + b·n(n+1)/2 + c·n

Example: For Tn = n² + n, a=1, b=1, c=0. Find S₅:

S₅ = 1·(5·6·11/6) + 1·(5·6/2) = 55 + 15 = 70

6. Practice Problems

  1. Find the nth term of 3, 8, 15, 24, 35, ...
  2. Given Tn = 2n² - n, find T₆.
  3. Find the sum of the first 6 terms of Tn = n² + 2n + 1.

7. Quiz: Test Your Understanding

1) Sequence: 5, 12, 21, 32, ... What is the second difference?



2) If Tn = n² + n, what is T₇?



3) The sum of the first 4 terms of Tn = 2n² - n is:



More Worked Examples — Quadratic Sequences

Here are additional examples to deepen your understanding of quadratic sequences and series. These examples include different patterns and contexts, ranging from simple numerical patterns to applied problems.

Example 1: Find the nth term of 4, 9, 16, 25, 36, ...

First differences: 5, 7, 9, 11 → Second differences: 2, 2, 2 (constant). Hence, it’s quadratic.

2a = 2 → a = 1. Assume Tn = n² + b n + c.

For n=1: 1 + b + c = 4 → b + c = 3

For n=2: 4 + 2b + c = 9 → 2b + c = 5

Subtract equations: b = 2, c = 1

Therefore, Tn = n² + 2n + 1 (a perfect square sequence).

Example 2: Find the nth term of 7, 15, 27, 43, 63, ...

First differences: 8, 12, 16, 20 → Second differences: 4, 4, 4 (constant)

2a = 4 → a = 2

Assume Tn = 2n² + b n + c

For n=1: 2 + b + c = 7 → b + c = 5

For n=2: 8 + 2b + c = 15 → 2b + c = 7

Subtract equations: b = 2, c = 3

Therefore, Tn = 2n² + 2n + 3

Example 3: Real-world application — Height of a thrown ball

A ball’s height at 1-second intervals is measured as follows (in meters): 0, 14, 24, 30, 32, 30, 24, 14, 0.

The pattern of heights forms a quadratic sequence because the second differences are constant.

By calculating differences, the formula can be modeled as: h(t) = -2t² + 16t.

Interpretation: The ball reaches its maximum height when t = 4 seconds, and h(4) = 32 m.

Example 4: Find the sum of the first 5 terms of Tn = 3n² - n + 2

a = 3, b = -1, c = 2.

Use formula Sn = a·n(n+1)(2n+1)/6 + b·n(n+1)/2 + c·n

S₅ = 3·(5·6·11)/6 + (-1)·(5·6)/2 + 2·5

S₅ = 165 - 15 + 10 = 160

Example 5: Find the nth term of 10, 17, 28, 43, 62, ...

First differences: 7, 11, 15, 19 → Second differences: 4, 4, 4 (constant)

2a = 4 → a = 2. Assume Tn = 2n² + b n + c

For n=1: 2 + b + c = 10 → b + c = 8

For n=2: 8 + 2b + c = 17 → 2b + c = 9

Subtract equations: b = 1, c = 7

Therefore, Tn = 2n² + n + 7

Challenge Quiz

1) The nth term of a sequence is Tn = n² + 3n + 2. What is T₁₀?



2) The first term of a quadratic sequence is 2 and the second term is 5. If the second difference is 3, what is the nth term?