A Different Way to Solve Quadratic Equation: Po Shen Loh’s Method

For years, students have been taught the same method to solve quadratic equations: the quadratic formula. But what if there was a different way—one that’s more intuitive and doesn’t require memorizing a complex formula? Enter Po-Shen Loh’s method, a groundbreaking approach that has revolutionized how we think about solving quadratic equations.

👨‍🔬 Who is Po-Shen Loh?

Po-Shen Loh is a renowned mathematician and professor at Carnegie Mellon University. He’s also the national coach of the USA Mathematical Olympiad team. In 2019, he introduced a method that challenges thousands of years of mathematical tradition by offering a more intuitive approach to solve quadratic equations.

🔍 The Traditional Method vs. Po-Shen Loh’s Method

Traditional Quadratic Formula

For the equation \(x^2 + bx + c = 0\), we use:

Po-Shen Loh’s Approach

Instead of memorizing a formula, this method uses the relationship between roots and completing the square in a more intuitive way.

📐 Understanding the Method

The general formula of quadratic equation is

where \(a \neq 0\). If the roots of quadratic equation are \(p\) and \(q\), we have relationship of \(p\) and \(q\) using Vieta’s Formula as follows:

Po-Shen Loh’s Key Insight

The brilliant insight is to express the roots in terms of their average and distance from the average:

Let \(m = -\frac{b}{2a}\) be the average of the roots. We can write the roots as:

• \[p = m + u\]
• \[q = m - u\]

where \(u\) is the distance from the average to each root.

Finding the Distance \(u\)

Since \(p \cdot q = \frac{c}{a}\), we substitute:

Hence,

We can choose \(u\) either in positive or negative. This will only affect \(p\) and \(q\) inversely.

For instance, we choose \(u\) in positive term. Therefore, the roots are:

• \[p = m + u\]
• \[q = m - u\]

where \(m = -\frac{b}{2a}\) and \(u = \sqrt{m^2 - \frac{c}{a}}\)

🔢 Example

Let’s solve \(2x^2 + 8x + 6 = 0\) using Po-Shen Loh’s method:

Step 1: Identify coefficients: \(a = 2\), \(b = 8\), \(c = 6\)

Step 2: Find the average of roots: \(m = -\frac{b}{2a} = -\frac{8}{2(2)} = -\frac{8}{4} = -2\)

Step 3: Calculate the distance \(u\): \(u = \sqrt{\left(-\frac{b}{2a}\right)^2 - \frac{c}{a}} = \sqrt{(-2)^2 - \frac{6}{2}} = \sqrt{4 - 3} = \sqrt{1} = 1\)

Step 4: Find the roots using \(p = m + u\) and \(q = m - u\):

• \[p = -2 + 1 = -1\]
• \[q = -2 - 1 = -3\]

Verification: \(2(x + 1)(x + 3) = 2(x^2 + 4x + 3) = 2x^2 + 8x + 6\) ✓

🎯 Comparing with Traditional Method

For \(2x^2 + 8x + 6 = 0\) using the quadratic formula:

So \(x = \frac{-8 + 4}{4} = -1\) or \(x = \frac{-8 - 4}{4} = -3\)

Same result! But Po-Shen Loh’s method provides geometric insight: the roots are symmetrically placed around their average \(m = -2\).

🎯 Conclusion

Po-Shen Loh’s method represents a conceptual breakthrough in mathematics education. Rather than memorizing formulas, students learn to:

• Think geometrically about algebraic problems
• Understand symmetry in quadratic equations
• Connect different mathematical concepts (algebra, geometry, Vieta’s formulas)

This method shows that even classical problems can be approached with fresh insights. It’s not just about finding roots—it’s about understanding the beautiful structure underlying quadratic equations.

Why This Matters

Mathematics isn’t just about computation; it’s about pattern recognition and conceptual understanding. Po-Shen Loh’s method exemplifies this philosophy, turning a mechanical process into an intuitive one.

What’s Next

Try the practice problems above, and you’ll discover that this method becomes second nature. More importantly, you’ll develop a geometric intuition for quadratic equations that will serve you well in higher mathematics.

Remember: The goal isn’t to replace the quadratic formula, but to understand quadratic equations more deeply. Sometimes, understanding is more valuable than memorization.

🎯 Why This Method is Revolutionary

1. Intuitive Understanding

Instead of memorizing a formula, students understand the geometric meaning of what they’re doing.

2. Visual Approach

The method naturally leads to visualizing roots as points equidistant from their average on the number line.

3. Connects Concepts

It beautifully connects:

• Vieta’s formulas
• Completing the square
• The geometric interpretation of roots

4. Reduces Memorization

Students don’t need to memorize the quadratic formula—they can derive the solution from basic principles.

🧮 Advantages of Po-Shen Loh’s Method

For Students

• Better conceptual understanding
• Less memorization required
• Visual and intuitive approach
• Connects multiple mathematical concepts

For Teachers

• Easier to explain the ‘why’ behind the method
• Helps students see patterns in mathematics
• Encourages mathematical thinking over rote memorization

Computational Efficiency

While conceptually superior, the traditional formula might be faster for computational purposes, especially with coefficients that don’t simplify nicely.

🎓 Educational Impact

This method represents a shift in mathematical education from:

• Memorization → Understanding
• Formula application → Conceptual thinking
• Procedural knowledge → Conceptual knowledge

📚 Conclusion

Po-Shen Loh’s method isn’t just another way to solve quadratic equations—it’s a paradigm shift that emphasizes understanding over memorization. While the traditional quadratic formula will always have its place, this new approach offers students a more intuitive path to mastering one of algebra’s most fundamental concepts.

As mathematics education continues to evolve, methods like this remind us that there’s always room for innovation, even in topics that have been taught the same way for millennia.

“The goal is not just to solve equations, but to understand the beautiful relationships that make the solutions possible.” - Po-Shen Loh

Have you tried this method? Share your thoughts and experiences in the comments below!