Question

При каких значениях параметра p квадратное уравнение 2x^2-7x+3p=0 имеет только один корень?

Answer 1

$2x^2-7x+3p=0\\ D=7^2-4*2*3p=0\\ 49-24p=0\\ -24p=-49\\ p=\frac{49}{24}$

Answer 2

2x^2-7x+3p=0
D=49-24p=0
49-24p=0
p=49/24

Answer 3

Quadratic Equation with One Root

To find the values of the parameter p for which the quadratic equation 2x^2 - 7x + 3p = 0 has only one root, we can use the discriminant of the quadratic equation. The discriminant is given by the formula Δ = b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0.

For a quadratic equation to have only one root, the discriminant must be equal to zero. This condition is given by the equation Δ = 0.

Calculation of Discriminant

In the given quadratic equation 2x^2 - 7x + 3p = 0, the coefficients are: - a = 2 - b = -7 - c = 3p

Using the formula for the discriminant, we have: Δ = (-7)^2 - 4 * 2 * 3p

Solving for Δ, we get: Δ = 49 - 24p

Condition for One Root

For the quadratic equation to have only one root, the discriminant Δ must be equal to zero. Therefore, we have the equation: 49 - 24p = 0

Solving for p, we find: p = 49 / 24

So, the value of the parameter p for which the quadratic equation 2x^2 - 7x + 3p = 0 has only one root is p = 49 / 24.

[[SOURCE #5]]