Which of the following equations has roots that are both negative? (2022)

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Question: Which of the following equations has roots that are both negative? (2022)

Options:

Correct Answer: x² + 4x + 4 = 0

Exam Year: 2022

Solution:

The equation x² + 4x + 4 = 0 has roots -2 and -2, which are both negative.

Which of the following equations has roots that are both negative? (2022)

Which of the following equations has roots that are both negative? (2022)

Step 1: Identify the equations you need to analyze for their roots.
Step 2: Recall that the roots of a quadratic equation in the form ax² + bx + c = 0 can be found using the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a).
Step 3: For each equation, calculate the discriminant (b² - 4ac) to determine the nature of the roots.
Step 4: If the discriminant is positive, there are two distinct roots. If it is zero, there is one repeated root. If it is negative, there are no real roots.
Step 5: Check the sign of the roots using the quadratic formula. If both roots are negative, then the equation has roots that are both negative.
Step 6: For the equation x² + 4x + 4 = 0, substitute a = 1, b = 4, and c = 4 into the quadratic formula.
Step 7: Calculate the discriminant: 4² - 4(1)(4) = 16 - 16 = 0.
Step 8: Since the discriminant is 0, there is one repeated root: x = (-4 ± √0) / (2*1) = -2.
Step 9: Conclude that the equation x² + 4x + 4 = 0 has roots -2 and -2, which are both negative.

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