Question

Can someone please help me

Answer 1

y = ax^2 + 10x + 1

For our graph, y = 0 whenever it crosses the x-axis, so we must solve:

0 = ax^2 + 10x + 1

Using the quadratic formula:

[-10 ± sqrt(100 - 4a)]/(2a)

In order to have two distinct roots, our square root part of the answer above must be a real number that is not zero.  In other words:

100 - 4a > 0

Because if the value underneath the square root symbol is negative, we will have no solution, and if the value underneath the square root symbol is zero, then we will only get one solution.

So solve:

100 - 4a > 0

100 > 4a

25 > a

So a must have a value lower than 25.  Also note that when a = 0, we have a linear equation which only has one solution that crosses the x-axis, so we must exclude a = 0 as well.

So the answer is:

a < 25, where a ≠ 0

Answer 2

a < 25

Consider the discriminant Δ = b² - 4ac

If b² - 4ac > 0 , then 2 real and distinct roots

with a = a, b = 10 and c = 1

b² - 4a = 100 - 4a, thus

100 - 4a > 0 ( subtract 100 from both sides )

- 4a > - 100 ( divide both sides by - 4 )

remembering to reverse the inequality symbol from > to <

a < 25 for 2 distinct solutions (a ≠ 0 )