Question

Solve the following quadratic equation:

`x^2 - 3sqrt(5)x + 10 = 0`

Answer 1

1. Identify the coefficients

Compare the given equation with the standard quadratic form ax² + bx + c = 0.

a = 1

b = `-3sqrt(5)`

c = 10

2. Calculate the discriminant

The discriminant (D) determines the nature of the roots and is given by the formula:

D = b² – 4ac

Substitute the values of a, b and c:

`D = (-3sqrt(5)^2) - 4(1)(10)`

D = (9 × 5) – 40

D = 45 – 40

D = 5

3. Apply quadratic formula

The quadratic formula used to find the roots is:

`x = (-b ± sqrt(D))/(2a)`

Substitute the values of b, D and a into the formula:

`x = (-(-3sqrt(5)) +- sqrt(5))/(2(1))`

`x = (3sqrt(5) +- sqrt(5))/(2)`

4. Simplify the roots

Separate the equation into two cases to find both individual values of x:

Case 1 (Addition):

`x = (3sqrt(5) + sqrt(5))/2`

= `(4sqrt(5))/2`

= `2sqrt(5)`

Case 2 (Subtraction):

`x = (3sqrt(5) - sqrt(5))/2`

= `(2sqrt(5))/2`

= `sqrt(5)`