What is the value of k if the quadratic equation x^2 + kx + 16 = 0 has roots tha
Practice Questions
Q1
What is the value of k if the quadratic equation x^2 + kx + 16 = 0 has roots that are both negative? (2019)
-8
-7
-6
-5
Questions & Step-by-Step Solutions
What is the value of k if the quadratic equation x^2 + kx + 16 = 0 has roots that are both negative? (2019)
Step 1: Understand that we have a quadratic equation in the form of x^2 + kx + 16 = 0.
Step 2: Recall that for a quadratic equation ax^2 + bx + c = 0, the roots can be found using the formula: roots = (-b ± √(b² - 4ac)) / (2a).
Step 3: In our equation, a = 1, b = k, and c = 16.
Step 4: For the roots to be negative, the sum of the roots (which is -b/a = -k) must be negative. This means k must be positive.
Step 5: The product of the roots (which is c/a = 16) must also be positive. Since both roots are negative, their product is positive.
Step 6: To ensure both roots are negative, we also need to check the condition that the sum of the roots is less than 0, which means k must be negative.
Step 7: We can use Vieta's formulas: if the roots are r1 and r2, then r1 + r2 = -k and r1 * r2 = 16.
Step 8: Since r1 * r2 = 16 and both roots are negative, we can say that |r1| and |r2| must be greater than 4 (since the product of two negative numbers is positive).
Step 9: Therefore, the sum of the absolute values of the roots must be greater than 8, which means |k| > 8.
Step 10: Since k must be negative, we conclude that k = -8.
