If the quadratic equation x^2 + kx + 16 = 0 has roots that are both real and dis

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Question: If the quadratic equation x^2 + kx + 16 = 0 has roots that are both real and distinct, what is the condition for k? (2022)

Options:

Correct Answer: k < -8

Exam Year: 2022

Solution:

The discriminant must be positive: k^2 - 4*1*16 > 0, which simplifies to k^2 > 64, hence k < -8 or k > 8.

If the quadratic equation x^2 + kx + 16 = 0 has roots that are both real and distinct, what is the condition for k? (2022)

If the quadratic equation x^2 + kx + 16 = 0 has roots that are both real and distinct, what is the condition for k? (2022)

Step 1: Identify the quadratic equation given, which is x^2 + kx + 16 = 0.
Step 2: Recall that for a quadratic equation ax^2 + bx + c = 0, the discriminant (D) is calculated using the formula D = b^2 - 4ac.
Step 3: In our equation, a = 1, b = k, and c = 16.
Step 4: Substitute the values into the discriminant formula: D = k^2 - 4*1*16.
Step 5: Simplify the expression: D = k^2 - 64.
Step 6: For the roots to be real and distinct, the discriminant must be greater than zero: k^2 - 64 > 0.
Step 7: Rearrange the inequality: k^2 > 64.
Step 8: Solve for k by taking the square root of both sides: k < -8 or k > 8.

Discriminant of a Quadratic Equation – The discriminant (D) of a quadratic equation ax^2 + bx + c = 0 is given by D = b^2 - 4ac. It determines the nature of the roots: if D > 0, the roots are real and distinct; if D = 0, the roots are real and equal; if D < 0, the roots are complex.
Conditions for Real Roots – For the roots of a quadratic equation to be real and distinct, the discriminant must be greater than zero.