If the quadratic equation x^2 + kx + 16 = 0 has real roots, what is the conditio
Practice Questions
Q1
If the quadratic equation x^2 + kx + 16 = 0 has real roots, what is the condition on k?
k^2 >= 64
k^2 < 64
k > 16
k < 16
Questions & Step-by-Step Solutions
If the quadratic equation x^2 + kx + 16 = 0 has real roots, what is the condition on k?
Step 1: Identify the quadratic equation, which is x^2 + kx + 16 = 0.
Step 2: Recall that for a quadratic equation ax^2 + bx + c = 0, the discriminant is given by 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 quadratic equation to have real roots, the discriminant must be greater than or equal to zero: k^2 - 64 >= 0.
Step 7: Rearrange the inequality: k^2 >= 64.
Step 8: This means k can be either greater than or equal to 8 or less than or equal to -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. For the equation to have real roots, D must be greater than or equal to zero.
Conditions for Real Roots – The condition for a quadratic equation to have real roots is that the discriminant must be non-negative (D >= 0).
