# 3x^2 + 2x - k = 0, find 3α^2 - 2β in terms of k

• tony24810
In summary, the problem is asking for the value of 3α^2 - 2β in terms of k, given that α and β are the roots of the equation 3x^2 + 2x - k = 0. After some attempts, the solution is found to be 4/3 + k, which can be obtained by solving the equations αβ = -k/3 and α + β = -2/3 and substituting the results into the first equation. The work is shown in detail in the conversation.
tony24810

## Homework Statement

Let k be a constant. If α and β are the roots of the equation 3x^2 + 2x - k = 0, find the value of 3α^2 - 2β in terms of k.

## The Attempt at a Solution

Obviously, the usual

αβ = -k/3
α + β = -2/3

has been written but I couldn't put them into the equation required despite a full hour's effort.

Also, tried writing (-b+-sqrt(b^2-4ac))/2a, and put respective roots into the equation, it yields something similar to the provided solution, but has an extra root term.

The solution is 4/3 + k.

The latter method gets 4/3 + k + sqrt(4+12k)/6.

tony24810 said:

## Homework Statement

Let k be a constant. If α and β are the roots of the equation 3x^2 + 2x - k = 0, find the value of 3α^2 - 2β in terms of k.

## The Attempt at a Solution

Obviously, the usual

αβ = -k/3
α + β = -2/3

has been written but I couldn't put them into the equation required despite a full hour's effort.

Also, tried writing (-b+-sqrt(b^2-4ac))/2a, and put respective roots into the equation, it yields something similar to the provided solution, but has an extra root term.

The solution is 4/3 + k.

The latter method gets 4/3 + k + sqrt(4+12k)/6.
Considering k to be a constant, solve the second equation below for α or β, then substitute into the first equation.
αβ = -k/3
α + β = -2/3

Mark44 said:
Considering k to be a constant, solve the second equation below for α or β, then substitute into the first equation.
αβ = -k/3
α + β = -2/3

It has a β^2 term leftover.

tony24810 said:

## Homework Statement

Let k be a constant. If α and β are the roots of the equation 3x^2 + 2x - k = 0, find the value of 3α^2 - 2β in terms of k.

## The Attempt at a Solution

Obviously, the usual

αβ = -k/3
α + β = -2/3

has been written but I couldn't put them into the equation required despite a full hour's effort.

Also, tried writing (-b+-sqrt(b^2-4ac))/2a, and put respective roots into the equation, it yields something similar to the provided solution, but has an extra root term.

The solution is 4/3 + k.

The latter method gets 4/3 + k + sqrt(4+12k)/6.

ehild

haha omg i got it

got it

i tried again this time giving each equation their names, suddenly spot the identity that i didn't see before. hahahaha

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## 1. What is the purpose of this equation?

The purpose of this equation is to find the value of 3α^2 - 2β in terms of k, given that 3x^2 + 2x - k = 0.

## 2. How do I solve this equation?

To solve this equation, you can use the quadratic formula (-b ± √(b^2 - 4ac)) / 2a, where a = 3, b = 2, and c = -k. Plug in these values and solve for x to find the value of α and β. Then, substitute those values into 3α^2 - 2β to find the answer in terms of k.

## 3. Why is it important to find 3α^2 - 2β in terms of k?

Finding 3α^2 - 2β in terms of k allows us to understand the relationship between the variables in the equation. It also helps us to generalize the solution for different values of k.

## 4. Can this equation be solved using other methods besides the quadratic formula?

Yes, this equation can also be solved by factoring or completing the square. However, the quadratic formula is the most efficient method for solving equations in the form ax^2 + bx + c = 0.

## 5. Are there any restrictions on the values of k that can be used in this equation?

Yes, there are restrictions on the values of k. For the equation to have a solution, the discriminant (b^2 - 4ac) must be greater than or equal to 0. This means that k must be within a certain range for the equation to have a real solution.

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