Quadratic Formula Worksheet & Examples (Step-by-Step Guide)

Quadratic equations form the foundation of many algebraic concepts in mathematics. A quadratic equation is any equation of the form:

ax² + bx + c = 0

Where a, b, and c are constants and a≠0, the quadratic formula offers a universal method to solve any quadratic equation. It is given by;

\[x=\dfrac{-b\pm \sqrt{b^{2}-4ac}}{2a}\]

The quadratic formula, upon which the algebraic field is founded, has a long and rich history, tracing itself back through the ancient civilizations. Its earliest known source lies with Brahmagupta, the renowned 7th-century Indian mathematician. Brahmagupta, in his influential work Brahmasphutasiddhanta, described solutions to quadratic equations but in spoken, rather than symbolic, terms. Brahmagupta needs to be counted among the first to provide definitions of positive and negative roots and, in so doing, provide the foundations for future developments within the algebraic field.

Centuries on, the Persian mathematician Muhammad ibn Musa al-Khwarizmi, also referred to as the father of algebra, built on Brahmagupta’s ideas in his 9th-century book Al-Kitab al-Mukhtasar fi Hisab al-Jabr wal-Muqabala. Al-Khwarizmi formulated systematic procedures for the solution of quadratic equations, categorizing them into their normal forms and solving them geometrically, a breakthrough in their day.

In the 11th century, Persian polymath Omar Khayyam created the theory by solving cubic and quadratic equations with geometric methods that employed conic sections. As brilliant as he was, he never formulated the modern equation, yet his work greatly evolved the theory of mathematical equations.

Presently, the quadratic formula, as we know it, is the result of these great historical efforts. It remains a vital tool in IGCSE Mathematics, enabling students to solve equations with precision and confidence.

In this post, we’ll explore a complete step-by-step guide to using the quadratic formula, provide seven worked examples, and include a downloadable quadratic formula worksheet for your practice with online answers.

Understanding the Quadratic Formula: A Quick Recap

Before we jump into examples, here’s what each term means in the quadratic formula:

Discriminant (D=b²−4ac):
Determines the nature of the roots.
- If D>0, two real and distinct solutions
- If D=0, one real repeated root
- If D<0, complex roots
“±” sign:
Indicates there are two possible values for ‘x’, due to the square root.

Let’s now solve progressively challenging problems using this formula.

Solved Examples Of Quadratic Formula: Beginner to Advanced

🔹 Example 1: Basic integer roots

Solve:

x² − 5x + 6 = 0

Solution:

a = 1, b = −5, c = 6

\[\begin{aligned}x=\dfrac{-\left( -5\right) \pm \sqrt{\left( -5\right) ^{2}-4\left( 1\right) \left( 6\right) }}{2\left( 1\right) }=\dfrac{5\pm \sqrt{25-24}}{2}=\dfrac{5\pm \sqrt{1}}{2}=\dfrac{5\pm 1}{2}\end{aligned}\]

x = 3 , and x = 2

✅ Two distinct real roots.

🔹 Example 2: Perfect square trinomial

Solve:

x² + 4x + 4 = 0

Solution:

a = 1, b = 4, c = 4

\[ x = \frac{-4 \pm \sqrt{16 – 16}}{2} = \frac{-4}{2} = -2 \]

✅ One real repeated root.

🔹 Example 3: Coefficients with fractions

Solve:

2x²−3x−2.5=0

Solution:

a = 2, b = −3, c = −2.5

\[ x = \frac{3 \pm \sqrt{9 + 40}}{4} = \frac{3 \pm \sqrt{49}}{4} = \frac{3 \pm 7}{4} \] \[ x = \frac{10}{4} = 2.5 \quad \text{or} \quad x = \frac{-4}{4} = -1 \]

✅ Two distinct real roots, one is a decimal.

🔹 Example 4: Discriminant is negative (complex roots)

Solve:

x² + 2x + 5 = 0

Solution:

a = 1, b = 2, c = 5

\[x = \frac{-2 \pm \sqrt{4 – 20}}{2} = \frac{-2 \pm \sqrt{-16}}{2} \\ x = \frac{-2 \pm 4i}{2} = -1 \pm 2i\]

✅ Two complex roots.

🔹 Example 5: Non-factorable quadratic

Solve:

3x² + 4x + 2 = 0

Solution:

a = 3, b = 4, c = 2

\[x = \frac{-4 \pm \sqrt{16 – 24}}{6} = \frac{-4 \pm \sqrt{-8}}{6} = \frac{-4 \pm 2\sqrt{2}i}{6} \\ x = \frac{-2 \pm \sqrt{2}i}{3}\]

✅ Complex roots with surds.

🔹 Example 6: Large coefficients

Solve:

5x² − 120x + 700 = 0

Solution:

a = 5, b = −120, c = 700

\[x = \frac{120 \pm \sqrt{14400 – 14000}}{10} = \frac{120 \pm \sqrt{400}}{10} \\ x = \frac{120 \pm 20}{10} = 14 \text{ or } 10\]

✅ Clean integer roots even with large numbers.

🔹 Example 7: Advanced Surd Form

Solve:

x² − x − 1 = 0

Solution:

a = 1, b = −1, c = −1

\[x = \frac{1 \pm \sqrt{1 + 4}}{2} = \frac{1 \pm \sqrt{5}}{2}\]

✅ Irrational surd roots – Golden Ratio appears here!

📥 Downloadable Quadratic Formula Worksheet: For better Practice

We’ve prepared a downloadable worksheet for learners to test their skills. This worksheet includes:

✅ 20 MCQ-style and short questions

✅ Progressive difficulty (easy → challenging)

✅ Online answer key (no full solutions)

👉 Click to Download the Quadratic Formula Worksheet (PDF)

✅ Final Thoughts: Ready to Master Quadratic Formula?

The quadratic formula is a must-know for every IGCSE Mathematics student. Whether you are just starting out or aiming for top grades, mastering this technique will serve you in algebra, graphing, and beyond.

✨ Want to get better at solving such problems with a teacher’s guidance?

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