Roots and Coefficients: Sum and Product

1. The Core Relationship: Sum and Product of Roots

For any quadratic equation in the standard form $a{x}^2+bx+c=0$, if its roots are $\alpha$ and $\beta$, we don't need to find $\alpha$ and $\beta$ individually to know their sum or product. These values are directly encoded in the coefficients $a$, $b$, and $c$.

To see how this relationship is derived, let's turn to a clear explanation.

The derivation you just read leads to two fundamental formulas. Let's formalize them.

For a quadratic equation $a{x}^2+bx+c=0$ with roots $\alpha$ and $\beta$:

• Sum of Roots: $\alpha +\beta =-\frac{b}{a}$
• Product of Roots: $\alpha \beta =\frac{c}{a}$

This visual summary is a great reference to keep in mind.

Let's see this in action. For the equation $3x^2-7x-11=0$:

• $a=3$, $b=-7$, $c=-11$
• Sum of roots = $-\frac{b}{a}=-\frac{(-7)}{3}=\frac{7}{3}$
• Product of roots = $\frac{c}{a}=\frac{-11}{3}$

Notice how we found the sum and product in seconds, without using the quadratic formula.

2. Building Equations from Roots

This relationship is a two-way street. If you know the roots of a quadratic equation, you can construct the equation itself.

If the roots are $\alpha$ and $\beta$, then:

• Sum (S) = $\alpha +\beta$
• Product (P) = $\alpha \beta$

The equation can be written as:
$x^2-(\text{Sum of roots})x+(\text{Product of roots})=0$
$x^2-Sx+P=0$

Example: Find the quadratic equation whose roots are 4 and -7.

1. Sum (S): $4+(-7)=-3$
2. Product (P): $4\times (-7)=-28$
3. Equation: $x^2-(-3)x+(-28)=0⟹x^2+3x-28=0$

This method works for all types of roots—integers, fractions, and even irrational or complex numbers.

3. Solving Problems with Symmetric Functions of Roots

This is where the concept truly becomes a powerful problem-solving tool for exams like the CAT. You'll often be asked to find the value of an expression involving the roots (e.g., ${\alpha }^2+{\beta }^2$), without being expected to find the roots themselves. The key is to algebraically manipulate the expression until it's in terms of $(\alpha +\beta )$ and $(\alpha \beta )$.

Let's look at some common identities:

• ${\alpha }^2+{\beta }^2=(\alpha +\beta {)}^2-2\alpha \beta$
• $\frac{1}{\alpha }+\frac{1}{\beta }=\frac{\alpha +\beta }{\alpha \beta }$
• $(\alpha -\beta {)}^2=(\alpha +\beta {)}^2-4\alpha \beta$
• ${\alpha }^3+{\beta }^3=(\alpha +\beta {)}^3-3\alpha \beta (\alpha +\beta )$

Knowing these identities is crucial. Let's see them applied.

Let's walk through the logic of a typical problem.

Problem: If $\alpha$ and $\beta$ are the roots of $x^2-6x+4=0$, what is the value of ${\alpha }^2+{\beta }^2$?

1. Find Sum and Product:

   • Sum ($\alpha +\beta$) = $-\frac{-6}{1}=6$
   • Product ($\alpha \beta$) = $\frac{4}{1}=4$

2. Use the Identity:

   • We know ${\alpha }^2+{\beta }^2=(\alpha +\beta {)}^2-2\alpha \beta$.

3. Substitute and Solve:

   • ${\alpha }^2+{\beta }^2=(6{)}^2-2(4)=36-8=28$.

The answer is 28. We found it without ever knowing the actual values of $\alpha$ and $\beta$.

4. Finding Unknown Coefficients

Another common question type involves finding an unknown coefficient when you are given some information about the roots.

Example 1: In the equation $x^2+kx-54=0$, one root is 6. Find the other root and the value of $k$.

1. Use the Product of Roots: Let the roots be $\alpha =6$ and $\beta$.

   • Product ($\alpha \beta$) = $c/a=-54/1=-54$.
   • $6\times \beta =-54⟹\beta =-9$. The other root is -9.

2. Use the Sum of Roots:

   • Sum ($\alpha +\beta$) = $-b/a=-k/1=-k$.
   • $6+(-9)=-k⟹-3=-k⟹k=3$.

Example 2 (More advanced): For the equation $x^2-7x+k=0$, the difference between the roots is 3. Find $k$.

1. Find Sum and Product: Let the roots be $\alpha$ and $\beta$.

   • $\alpha +\beta =7$
   • $\alpha \beta =k$

2. Use the Given Information and an Identity:

   • We are given $\alpha -\beta =3$.
   • Use the identity: $(\alpha -\beta {)}^2=(\alpha +\beta {)}^2-4\alpha \beta$.

3. Substitute and Solve for k:

   • $(3{)}^2=(7{)}^2-4(k)$
   • $9=49-4k$
   • $4k=49-9=40$
   • $k=10$.

Conclusion

You've now added another powerful set of shortcuts to your toolkit. While the discriminant tells you about the type of roots, the sum and product formulas give you actionable values to work with, often bypassing the need for complex calculations. For the CAT, where every second counts, mastering these techniques is not just helpful—it's essential.

Key Takeaways:

• For any quadratic equation $a{x}^2+bx+c=0$, the sum of roots is $-b/a$ and the product of roots is $c/a$.
• You can form a quadratic equation if you know its roots ($\alpha ,\beta$) using the formula $x^2-(\alpha +\beta )x+\alpha \beta =0$.
• Complex expressions involving roots (like ${\alpha }^2+{\beta }^2$) can be simplified by expressing them in terms of the sum and product of roots.
• These relationships are frequently used to find unknown coefficients in an equation based on given properties of the roots.

Preview of the Next Lesson:
We will now move from equations (finding specific values) to inequalities (finding a range of values). In the next lesson, we will cover how to solve linear and quadratic inequalities, introducing a systematic technique known as the wavy curve method, which is invaluable for quickly solving complex inequality problems.