Question

If α and β are roots of the quadratic equation x² – 7x + 10 = 0, find the quadratic equation whose roots are α² and β².

Answer 1

Given α and β are roots of x² – 7x + 10 = 0

∴ α + β = `-((-7))/1` = 7  ...(1)

and αβ = `10/1` = 10  ...(2)

∴ (α + β)² = α² + β² + 2αβ

∴ (7)² = α² + β² + 2(10)  ...(from (1) and (2))

`\implies` α² + β² = 49 – 20 = 29  ...(3)

Now, the quadratic equation whose roots are α² and β² will be

x² – (α² + β²) = x + α²β² = 0

`\implies` x² – 29x + (10)² = 0  ...(from (2) and (3))

`\implies` x² – 29x + 100 = 0