The quadratic equation 1.5x^2 - kx + 6 = 0, where k is a positive constant, has exactly on… — ACT

The quadratic equation $1.5x^2 - kx + 6 = 0$ , where $k$ is a positive constant, has exactly one real solution. What is the value of $k$ ?

Answer: 6

For the quadratic equation

1.5x^2 - kx + 6 = 0

to have exactly one real solution, the discriminant

b^2 - 4ac

must equal

0

. Substituting

a = 1.5

b = -k

, and

c = 6

gives

(-k)^2 - 4(1.5)(6) = k^2 - 36 = 0

, which yields

k^2 = 36

. Since

k

must be a positive constant,

k

must be

6

Identify the coefficients of the quadratic equation

1.5x^2 - kx + 6 = 0

a = 1.5

b = -k

, and

c = 6

To use the discriminant formula, we need to know the values of

a

b

, and

c

from the standard form

ax^2 + bx + c = 0

Set the discriminant equal to zero.

b^2 - 4ac = 0

A quadratic equation has exactly one real solution if and only if its discriminant is equal to zero.

Substitute the coefficients into the discriminant formula and simplify.

k^2 - 36 = 0

Substituting

a = 1.5

b = -k

, and

c = 6

into the formula gives

(-k)^2 - 4(1.5)(6) = k^2 - 36 = 0

Solve the equation for the positive constant

k

k = 6

Solving

k^2 = 36

gives

k = 6

k = -6

. Since the problem states that

k

is a positive constant, we choose

k = 6

Determining the number of real solutions of a quadratic equation using the discriminant

Question