monir301 wrote:

If a > 0 and b < 0, which of the following statements are true about the values of x that solve the equation x² - ax + b = 0 ?

Indicate all such statements.

A. They have opposite signs.

B. Their sum is greater than Zero.

C. Their product equals -b

If we were to FACTOR x^2 - ax + b into the form (x + k)(x + j), we'd have two clues to help us.

First, we know that the product jk must equal b.

Since we're told that b is NEGATIVE, we know that one of the variables (j or k) is POSITIVE, and the other is NEGATIVE.

Second, we know that the sum, j + k, is equal to -a

Since a is positive, we know that -a is NEGATIVE

So, we already know that one of the variables (j or k) is POSITIVE, and the other is NEGATIVE, AND we now know that their SUM is negative.

This means that the negative value (j or k) has a greater magnitude than the positive value. This is the only way that the sum, j + k, can be negative.

So, here are some possible expressions/equations that meet these criteria:

1. x² - 3x - 15 = 0

2. x² - 2x - 8 = 0

3. x² - 5x - 6 = 0

4. x² - 10x - 39 = 0

When we factor and solve them, we get:

1. (x - 5)(x + 3) = 0, so x = 5 or -3

2. (x - 4)(x + 2) = 0, so x = 4 or -2

3. (x - 6)(x + 1) = 0, so x = 6 or -1

4. (x - 13)(x + 3) = 0, so x = 13 or -3

Let's examine the 3 statements:

A. They have opposite signs.

This is true.

B. Their sum is greater than Zero.

This is true.

C. Their product equals -b

This LOOKS true, but it isn't.

For example, in the first equation x² - 3x - 15 = 0, the value of b is -15, which means b = 15

When we solved the equations, we found that x = 5 or -3

So, the product of the solutions is -15 (not 15)

Cheers,

Brent

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Brent Hanneson – Creator of greenlighttestprep.com

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