Determine the exact value of k so that the quadratic function f(x) = x2 - kx + 5 has only one zero.
to have one root, (actually two equal roots)
b^2 - 4ac = 0
k^2 - 4(1)(5) = 0
k^2 = 20
k = ± √20 = ± 2√5
check:
x^2 + 2√5x + 5 = (x + √5)^2
x^2 - 2√5x + 5) = (x - √5)^2
To find the value of k so that the quadratic function has only one zero, we need to analyze the discriminant of the quadratic equation.
The discriminant is given by the formula:
Δ = b^2 - 4ac
In the given quadratic function f(x) = x^2 - kx + 5, the coefficients are:
a = 1 (coefficient of x^2)
b = -k (coefficient of x)
c = 5
For a quadratic function to have only one zero, the discriminant must be equal to zero. This is because the quadratic formula, when the discriminant is zero, will produce only one solution.
Therefore, we set the discriminant Δ to zero:
Δ = b^2 - 4ac = 0
(-k)^2 - 4(1)(5) = 0
k^2 - 20 = 0
Rearranging the equation:
k^2 = 20
Taking the square root of both sides:
k = ±√20
Simplifying further:
k = ±2√5
So, the exact values of k for which the quadratic function f(x) = x^2 - kx + 5 has only one zero are ±2√5.