Forming a Quadratic Equation with Given Roots

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.

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

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

∴ 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.

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

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

, (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

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

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

(b) The required quadratic equation is

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.

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

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

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

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

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