Roots of quadratic equations

Introduction to Roots of Quadratic Equations

Welcome to the world of Further Pure Mathematics! In your previous studies, you’ve spent a lot of time solving quadratic equations to find the values of \(x\). In this chapter, we are going to flip the script. Instead of just finding the roots, we are going to explore the beautiful relationships between the roots and the coefficients of the equation.

Think of this like being a detective. Even if you don't know exactly who the "roots" are, you can find out a lot about their "personality" (their sum and their product) just by looking at the equation itself. This is a powerful tool that helps us solve complex problems and build new equations from scratch.

1. The Basics: Sum and Product of Roots

Every quadratic equation can be written in the standard form:
\(ax^2 + bx + c = 0\)

Let’s call the two roots of this equation \(\alpha\) (alpha) and \(\beta\) (beta). Even without using the quadratic formula to find them, we know two fundamental rules called Vieta’s Formulas:

• The Sum of the Roots: \(\alpha + \beta = -\frac{b}{a}\)
• The Product of the Roots: \(\alpha\beta = \frac{c}{a}\)

Did you know? These relationships work even if the roots are complex numbers or repeated! As long as you have a quadratic, these two "DNA markers" will always be true.

How to use this (Step-by-Step):

1. Ensure your equation is equal to zero.
2. Identify your values for \(a\), \(b\), and \(c\). Be very careful with negative signs!
3. Plug them into the formulas above.

Example: For the equation \(2x^2 - 8x + 5 = 0\):
\(a = 2, b = -8, c = 5\)
Sum (\(\alpha + \beta\)) \(= -(-8) / 2 = 8 / 2 = 4\)
Product (\(\alpha\beta\)) \(= 5 / 2 = 2.5\)

Quick Review Box:
Sum \(= -\frac{b}{a}\)
Product \(= \frac{c}{a}\)
Common Mistake: Forgetting the minus sign in the sum formula. Always remember: "Sum is the opposite of \(b\) over \(a\)."

2. Manipulating Expressions

Sometimes, exam questions won't ask for the sum or product directly. Instead, they might ask you to find the value of something like \(\alpha^2 + \beta^2\) or \(\alpha^3 + \beta^3\). Since we only know the values of \((\alpha + \beta)\) and \((\alpha\beta)\), we have to rewrite these tricky expressions using only those two building blocks.

Key Identities to Memorize:

The Square Identity:
\(\alpha^2 + \beta^2 \equiv (\alpha + \beta)^2 - 2\alpha\beta\)

The Cube Identity:
\(\alpha^3 + \beta^3 \equiv (\alpha + \beta)^3 - 3\alpha\beta(\alpha + \beta)\)

Don't worry if this seems tricky at first! You can prove the square identity by expanding \((\alpha + \beta)^2\). You’ll get \(\alpha^2 + 2\alpha\beta + \beta^2\). To leave just the squares, you simply subtract the \(2\alpha\beta\) from the middle!

Other Common Manipulations:

• Fractions: \(\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha\beta}\) (Just find a common denominator!)
• More Squares: \(\frac{1}{\alpha^2} + \frac{1}{\beta^2} = \frac{\alpha^2 + \beta^2}{(\alpha\beta)^2}\)

Key Takeaway: Before calculating anything, always try to rewrite the expression so it only contains \((\alpha + \beta)\) and \(\alpha\beta\).

3. Forming New Equations

A very common exam task is to create a new quadratic equation whose roots are related to the old ones (for example, the new roots might be \(\alpha^3\) and \(\beta^3\)).

To do this, use this template for a quadratic equation where the coefficient of \(x^2\) is 1:
\(x^2 - (\text{Sum of New Roots})x + (\text{Product of New Roots}) = 0\)

Step-by-Step Process:

1. Find the sum and product of the original roots (\(\alpha + \beta\) and \(\alpha\beta\)).
2. Calculate the New Sum (add the two new roots together).
3. Calculate the New Product (multiply the two new roots together).
4. Substitute these into the template: \(x^2 - (\text{New Sum})x + (\text{New Product}) = 0\).
5. If the question asks for integer coefficients, multiply the whole equation by the denominator to clear any fractions.

Example: Find an equation with roots \(\frac{1}{\alpha}\) and \(\frac{1}{\beta}\).
New Sum \(= \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha\beta}\)
New Product \(= \frac{1}{\alpha} \times \frac{1}{\beta} = \frac{1}{\alpha\beta}\)
Now just plug in the numbers you found from the original equation!

Analogy: Think of this like a recipe. The original roots are your ingredients. The formulas in Section 2 are your tools to prep the ingredients. Finally, the template in Section 3 is the oven where you bake the new equation.

Summary of Key Points

• For \(ax^2 + bx + c = 0\), Sum \(= -b/a\) and Product \(= c/a\).
• Use Algebraic Identities to turn complex expressions into sums and products.
• To build a new equation, find the New Sum (S) and New Product (P) and use \(x^2 - Sx + P = 0\).
• Always double-check your signs, especially when \(b\) is already negative.

Practice Tip: The more you practice the \(\alpha^3 + \beta^3\) identity, the more natural it will feel. It's one of the most common "harder" marks in this section!