Find the value of k if the equation x² + kx + 16 = 0 has no real roots. (2022)

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Question: Find the value of k if the equation x² + kx + 16 = 0 has no real roots. (2022)

Options:

Correct Answer: k > 8

Solution:

For no real roots, the discriminant must be less than zero: k² - 4*1*16 < 0, which gives k > 8.

Find the value of k if the equation x² + kx + 16 = 0 has no real roots. (2022)

Find the value of k if the equation x² + kx + 16 = 0 has no real roots. (2022)

Step 1: Identify the equation given, which is x² + kx + 16 = 0.
Step 2: Understand that for a quadratic equation to have no real roots, the discriminant must be less than zero.
Step 3: Recall the formula for the discriminant, which is D = b² - 4ac, where a, b, and c are the coefficients from the equation ax² + bx + c.
Step 4: In our equation, a = 1, b = k, and c = 16.
Step 5: Substitute the values into the discriminant formula: D = k² - 4*1*16.
Step 6: Simplify the expression: D = k² - 64.
Step 7: Set up the inequality for no real roots: k² - 64 < 0.
Step 8: Rearrange the inequality: k² < 64.
Step 9: Take the square root of both sides: |k| < 8.
Step 10: This means k must be between -8 and 8, so k > 8 or k < -8.

Discriminant – The discriminant of a quadratic equation determines the nature of its roots; if it is less than zero, the equation has no real roots.
Quadratic Equation – A quadratic equation is in the form ax² + bx + c = 0, where a, b, and c are constants.
Inequalities – Understanding how to manipulate inequalities is crucial for determining the range of values for k.