For the quadratic equation 2x^2 - 4x + k = 0 to have real roots, what is the con
Practice Questions
Q1
For the quadratic equation 2x^2 - 4x + k = 0 to have real roots, what is the condition on k?
k >= 0
k <= 0
k >= 2
k <= 2
Questions & Step-by-Step Solutions
For the quadratic equation 2x^2 - 4x + k = 0 to have real roots, what is the condition on k?
Correct Answer: k <= 2
Step 1: Identify the quadratic equation, which is 2x^2 - 4x + k = 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 = 2, b = -4, and c = k.
Step 4: Substitute the values of a, b, and c into the discriminant formula: D = (-4)^2 - 4*2*k.
Step 5: Calculate (-4)^2, which is 16, so now we have D = 16 - 8k.
Step 6: For the quadratic to have real roots, the discriminant must be greater than or equal to zero: 16 - 8k >= 0.
Step 7: Rearrange the inequality: 16 >= 8k.
Step 8: Divide both sides by 8: 2 >= k.
Step 9: This can also be written as k <= 2.
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 non-negative (D >= 0).
Conditions for Real Roots – Understanding that for a quadratic equation to have real roots, the discriminant must be greater than or equal to zero.
