Question

It is known that x1 and x2 are roots of the equation 3x^2+2x+k=0, where 2x1=−3x2. Find k.

Answer 1

***k = -8***

Step-by-step explanation:

Since the roots of the equation 3x² + 2x + k are x₁ and x₂ and 2x₁ = 3x₂.

By the roots of an equation, ax² + bx + c , where its roots are x₁ and x₂. It follows that sum of roots are x₁ + x₂ = -b/a and product of roots are x₁x₂ = c/a.

Comparing both equations, a = 3 b = 2 and c = k.

So, x₁ + x₂ = -b/a = -2/3 and x₁x₂ = c/a = k/3

x₁ + x₂ = -2/3 and x₁x₂ = k/3

3x₁ + 3x₂ = -2 (1) and 3x₁x₂ = k (2)

Since 2x₁ = -3x₂, and -2x₁ = 3x₂substituting this into equations (1) and (2) above, we have

3x₁ + 3x₂ = -2 (1) and 3x₁x₂ = k (2)

3x₁ + (-2x₁) = -2 and x₁(3x₂) = k

3x₁ - 2x₁ = -2 and x₁(3x₂) = k

x₁ = -2 and x₁(-2x₁) = k

x₁ = -2 and -2x₁² = k

Substituting x₁ = -2 into -2x₁² = k, we have

-2x₁² = k

-2(-2)² = k

***-2(4) = k

-8 = k

So, k = -8
Note: the stars are too show the mistakes and the real answer

Answer 2

By applying the relationships of the sum and product of roots for quadratic equations, we find that the value of k in the equation 3x² + 2x + k = 0 is -8, considering 2x₁ = -3x₂.

Step-by-step explanation:

To find the value of k for the quadratic equation 3x² + 2x + k = 0, where 2x₁ = -3x₂, we can use the properties of the roots of a quadratic equation. We know that the sum of the roots (x₁ + x₂) is equal to -b/a and the product of the roots (x₁ × x₂) is equal to c/a for a quadratic equation ax² + bx + c = 0.

Given that 2x₁ = -3x₂, we can express x1 in terms of x₂: x₁ = (-3/2)x₂. By summing the roots we get:

x₁ + x₂ = (-3/2)x₂ + x₂ = (-1/2)x₂

Using the equation for the sum of the roots we have:

(-1/2)x₂ = -2/3

So, x₂ = (4/3) and x₁ = (-3/2)(4/3) = -2.

Using the product of roots we get:

x₁ × x₂ = k/3

-2 × (4/3) = k/3

k = -8. Therefore, the value of k in the quadratic equation is -8.