If the roots of the quadratic equation x² + px + q = 0 are both negative, which

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Question: If the roots of the quadratic equation x² + px + q = 0 are both negative, which of the following must be true?

Options:

Correct Answer: p > 0 and q > 0

Solution:

For both roots to be negative, the sum (p) must be positive and the product (q) must also be positive.

If the roots of the quadratic equation x² + px + q = 0 are both negative, which of the following must be true?

If the roots of the quadratic equation x² + px + q = 0 are both negative, which of the following must be true?

Step 1: Understand that a quadratic equation is in the form x² + px + q = 0.
Step 2: Identify that the roots of the equation are the values of x that make the equation equal to zero.
Step 3: Recall that the sum of the roots (let's call them r1 and r2) is given by -p (from the formula -b/a for the equation ax² + bx + c = 0).
Step 4: Since both roots r1 and r2 are negative, their sum (r1 + r2) must also be negative. Therefore, -p must be negative, which means p must be positive.
Step 5: Next, remember that the product of the roots is given by q (from the formula c/a).
Step 6: Since both roots are negative, their product (r1 * r2) will be positive. Therefore, q must also be positive.
Step 7: Conclude that for both roots to be negative, p must be positive and q must be positive.

Roots of Quadratic Equations – Understanding the relationship between the coefficients of a quadratic equation and the nature of its roots, specifically the conditions for the roots to be negative.
Vieta's Formulas – Applying Vieta's formulas, which relate the coefficients of a polynomial to sums and products of its roots.