If the quadratic equation x^2 + kx + 9 = 0 has no real roots, what is the condit
Practice Questions
Q1
If the quadratic equation x^2 + kx + 9 = 0 has no real roots, what is the condition on k?
k < 6
k > 6
k < 0
k > 0
Questions & Step-by-Step Solutions
If the quadratic equation x^2 + kx + 9 = 0 has no real roots, what is the condition on k?
Step 1: Identify the quadratic equation given, which is x^2 + kx + 9 = 0.
Step 2: Recall that a quadratic equation has no real roots if its discriminant is less than zero.
Step 3: The discriminant (D) for the equation ax^2 + bx + c = 0 is given by the formula D = b^2 - 4ac.
Step 4: In our equation, a = 1, b = k, and c = 9.
Step 5: Substitute the values into the discriminant formula: D = k^2 - 4*1*9.
Step 6: Simplify the expression: D = k^2 - 36.
Step 7: Set the discriminant less than zero for no real roots: k^2 - 36 < 0.
Step 8: Rearrange the inequality: k^2 < 36.
Step 9: Take the square root of both sides: |k| < 6.
Step 10: This means k must be between -6 and 6 for the quadratic equation to have no real roots.
Quadratic Equations – Understanding the properties of quadratic equations, particularly the role of the discriminant in determining the nature of the roots.
Discriminant – The discriminant (b^2 - 4ac) helps to determine whether a quadratic equation has real roots, complex roots, or repeated roots.
Inequalities – Solving inequalities to find conditions on the variable k that ensure the quadratic has no real roots.
