Question

Decide whether this statement is always true, never true, or sometimes true, and explain why: In a quadratic equation of the form 0 = ax2 + bx + c, there will be two real solutions if a and c are of opposite signs

Answer 1

Always true

Explanation:

Here, we want to evaluate if the given statement is true

The discriminant D can be calculated using the formula;

b^2 - 4ac

if D is greater than 0, then we will have only real roots

In a case where a and c are of opposite signs;

then -4ac will always give a positive value which can be added to b^2

This positive value will ensure that D is greater than zero and hence we can have only real roots

So this assertion is always true if a and c are of opposite signs