1. Forming a Quadratic Equation with Given Roots

2. Miss Chan, I have learnt how to solve a quadratic equation, but how can I form a quadratic equation from two given roots?
We can form the equation by reversing the process of solving a quadratic equation by the factor method.

3. Consider the following example:
Solve an equation
x2 – 4x + 3 = 0
(x – 1)(x – 3) = 0
x – 1 = 0 or x – 3 = 0
x = 1 or x = 3

4. Form an equation from two given roots
Consider the following example:
Form an equation from two given roots
(x – 1)(x – 3) = 0
Reversing the process
x – 1 = 0 or x – 3 = 0
x = 1 or x = 3

5. ∴ The quadratic equation is x2 – 4x + 3 = 0.
Now, let’s study the steps of forming a quadratic equation from given roots (1 and 3) again.
x = 1 or x = 3
Note this key step.
x – 1 = 0 or x – 3 = 0
(x – 1)(x – 3) = 0
x2 – 4x + 3 = 0
∴ The quadratic equation is x2 – 4x + 3 = 0.

6. In general,
Roots ,
(x – )(x – ) = 0
Quadratic Equation α β α β

7. For example,
Roots ,
(x – )(x – ) = 0
Quadratic Equation 1 2 1 2

8. , (x – )(x – ) = 0 –4 (x – )[x – ] = 0 α 1 (–4) β 2 For example, Roots
Quadratic Equation –4
Quadratic Equation
(x – )[x – ] = 0 α 1
(–4) β 2

9. In general,
If α and β are the roots of a quadratic equation in x, then the equation is:
(x – α)(x – β) = 0

10. Follow-up question
In each of the following, form a quadratic equation in x with the given roots, and write the equation in the general form.
(a) –1, –5 (b)
(a) The required quadratic equation is
[x – (–1)][x – (–5)] = 0
(x + 1)(x + 5) = 0
x2 + x + 5x + 5 = 0
x2 + 6x + 5 = 0

11. (b) The required quadratic equation is

12. but it is too tedious to expand the left hand side of the equation
I am trying to form a quadratic equation whose roots are and ,
In fact, there is another method to form a quadratic equation. It helps you form this quadratic equation.

13. Using Sum and Product of Roots
Suppose α and β are the roots of a quadratic equation.
Then, the equation can be written as:
(x – )(x – ) = 0
x2 – x – x +  = 0
By expansion
x2 – ( + )x +  = 0
Sum of roots
Product of roots

14. Using Sum and Product of Roots
Suppose α and β are the roots of a quadratic equation.
Then, the equation can be written as:
x2 – (sum of roots)x + product of roots = 0

15. Let’s find the quadratic equation whose roots are and .
∴ The required quadratic equation is
x2 – (sum of roots)x + product of roots = 0

16. Follow-up question
Form a quadratic equation in x whose roots are and (Write your answer in the general form.)

17. Follow-up question
Form a quadratic equation in x whose roots are and (Write your answer in the general form.)
∴ The required quadratic equation is