The statement that is necessarily true if the roots of the quadratic equation ax^2 + bx + c = 0 are opposite in sign, with the root having the greater absolute value being negative, is that "b 0".

Here's why:

1. The roots of a quadratic equation ax^2 + bx + c = 0 are given by the formula (-b ± √(b^2 - 4ac)) / 2a.

2. If the roots are opposite in sign, it means one root is positive and the other is negative.

3. The root with the greater absolute value being negative implies that the positive root is smaller in magnitude than the negative root.

4. The sum of the roots is given by -b/a and the product of the roots is given by c/a.

5. Since the sum of a positive number (the smaller root) and a negative number (the larger root in absolute value) is negative, -b/a must be negative. Therefore, b must be positive as a is always positive (since it's the coefficient of x^2).

6. Hence, the statement "b 0" is necessarily true.

Question

The statement that is necessarily true if the roots of the quadratic equation ax^2 + bx + c = 0 are opposite in sign, with the root having the greater absolute value being negative, is that "b > 0".

Here's why:

1. The roots of a quadratic equation ax^2 + bx + c = 0 are given by the formula (-b ± √(b^2 - 4ac)) / 2a.

2. If the roots are opposite in sign, it means one root is positive and the other is negative.

3. The root with the greater absolute value being negative implies that the positive root is smaller in magnitude than the negative root.

4. The sum of the roots is given by -b/a and the product of the roots is given by c/a.

5. Since the sum of a positive number (the smaller root) and a negative number (the larger root in absolute value) is negative, -b/a must be negative. Therefore, b must be positive as a is always positive (since it's the coefficient of x^2).

6. Hence, the statement "b > 0" is necessarily true.

Knowee AI · Accepted Answer

The statement that is necessarily true if the roots of the quadratic equation ax^2 + bx + c = 0 are opposite in sign, with the root having the greater absolute value being negative, is that "b > 0".

Here's why:

1. The roots of a quadratic equation ax^2 + bx + c = 0 are given by the formula (-b ± √(b^2 - 4ac)) / 2a.

2. If the roots are opposite in sign, it means one root is positive and the other is negative.

3. The root with the greater absolute value being negative implies that the positive root is smaller in magnitude than the negative root.

4. The sum of the roots is given by -b/a and the product of the roots is given by c/a.

5. Since the sum of a positive number (the smaller root) and a negative number (the larger root in absolute value) is negative, -b/a must be negative. Therefore, b must be positive as a is always positive (since it's the coefficient of x^2).

6. Hence, the statement "b > 0" is necessarily true.

For a quadratic equation ax2 + bx + c = 0, which of the following statements is necessarily true, if the roots of the equation are opposite in sign, the root with the greater absolute value being negative?

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