We are given that a² = a + c, where ac > 0. We need to determine which of the following statements must be true.

To begin, let's simplify the given equation:

a² - a - c = 0

We can solve this quadratic equation using the quadratic formula:

a = [1 ± sqrt(1 + 4c)] / 2

Since ac > 0, we know that either both a and c are positive or both are negative.

Now, let's consider each statement:

a > 0: This statement must be true, since the solutions to the quadratic formula are both positive.

a < c: This statement may or may not be true, depending on the value of c. If c is negative, then a < c. But if c is positive, then a > c.

0 < a < 1: This statement may or may not be true, depending on the value of c. If c is less than 3/4, then both solutions to the quadratic formula are between 0 and 1. But if c is greater than 3/4, then one solution is greater than 1 and the other is less than 0.

a > 1: This statement may or may not be true, depending on the value of c. If c is greater than 3, then both solutions to the quadratic formula are greater than 1. But if c is less than 3, then one solution is greater than 1 and the other is negative.

Therefore, the only statement that must be true is a > 0. The other statements may or may not be true, depending on the value of c.

Answer: a > 0

Can we solve it by substituting the c value in the ac> 0 inequality also?

It gives a clear a value in that case.

\(a^2 = a + c\)

AND

\(ac > 0\)


Now, \(a^2 = a + c\)\(= a^2-a=c\)

\(a(a-1)=c\)

This means that a and a-1 are consecutive integers such as 3 and 2

We do know also that ac > 0 and a and c are both positive or negative numbers

BUt in the original stem we had a^2 and this is a clue to know that a is + and therefore c is + as well

Now we do have all the pieces of information

A. a> 0 this is true because if we know that c is + also a must be positive

B. a < c this could be true

3*2=6 and 3 < 6 and this is true. however if we have 2*1=2 the condition is satisfied but 2(a) = 2 (c) So b can be or not true and we need a MUST be true

C. 0<a<1 suppose a is 1/2 then we would have 1/2-1 which would be negative and negative * positive is negative. This is not possible

D. a>1 yes because if a would 1 then a-1 would be zero and this is not possible

So A and D are the correct choices

As for your question I am not sure I got what you mean but I think is not feasible
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