Question

Determine whether the following quadratic equation has real roots.

5x² – 9x + 4 = 0

1. Give reasons for your answer.
2. If the equation has real roots, identify them.

Answer 1

Given: 5x² – 9x + 4 = 0.

Step-wise calculation:

1. Compare with ax² + bx + c = 0:

a = 5

b = –9

c = 4

2. Compute the discriminant D = b² – 4ac:

D = (–9)² – 4 × 5 × 4 

= 81 – 80

= 1

Recall: The sign of D determines whether roots are real, zero or positive → two equal, two distinct real or non‑real; see discriminant discussion.

3. Since D = 1 > 0, there are two distinct real roots.

4. Find the roots using `x = (-b ± sqrt(D))/(2a)`:

`x = (-(-9) ± sqrt(1))/(2 xx 5)`

= `(9 ± 1)/10`

Thus, `x_1 = (9 + 1)/10`

= `10/10`

= 1

And `x_2 = (9 - 1)/10`

= `8/10`

= `4/5`

The equation has two distinct real roots because D = 1 > 0, namely x = 1 and x = `4/5`.