Question

Solve the following quadratic equations by completing the square method.

1. x² + x – 20 = 0 

2. x² + 2x – 5 = 0 

3. m² – 5m = -3

Answer 1

1. x² + x – 20 = 0 

If x² + x + k = (x + a)² , then 

x² + x + k = x² + 2ax + a² 

Comparing the coefficients, we get 

1 = 2a and k = a² 

∴ The roots of the given quadratic equation are 4 and -5.

2. x² + 2x – 5 = 0 

If x² + 2x + k = (x + a)² , then 

x² + 2x + k = x² + 2ax + a² 

Comparing the coefficients, we get 

2 = 2a and k = a² 

∴ a = 1 and k = (1)² = 1 

Now, x² + 2x – 5 = 0 

∴ x² + 2x + 1 – 1 – 5 = 0 

∴ (x + 1)² – 6 = 0 

∴ (x + 1)² = 6 

Taking square root of both sides, we get 

x + 1 = ± √6 

∴ x + 1 √6 or x + 1 = √6 

∴ x = √6 – 1 or x = -√6 – 1 

∴ The roots of the given quadratic equation are √6 -1 and – √6 -1.

3. m² – 5m = -3 

∴ m – 5m + 3 = 0 

If m² – 5m + k = (m + a)² , then 

m² – 5m + k = m² + 2am + a² 

Comparing the coefficients, we get 

-5 = 2a and k = a²