# Solve the quadratic equation that involves sum and product

• chwala
In summary, the conversation discussed various questions related to quadratic equations and their solutions. The participants discussed the roots of a given quadratic equation and how they relate to the values of α and β. They also discussed the values of α^3+β^3 and α^2+β^2, as well as the equation x^2-\dfrac{α^3+β^3}{αβ}x+αβ=0 and its solution. Overall, the conversation focused on finding solutions and understanding the relationships between different elements of quadratic equations.
chwala
Gold Member
Homework Statement
see attached
Relevant Equations
I am refreshing on this...Have to read broadly...i will start with (b) then i may be interested in alternative approach or any correction that may arise from my working. Cheers.

Kindly note that i do not have the solutions to the following questions...

For (b), we know that,
say, if ##x=α## and ##x=β## are roots of the given quadratic equation, then it follows that,
##-(α+β)=3.5##
##αβ=2##,
##(α-β)^2=(α+β)^2-4αβ##
##(α-β)^2=(3.5)^2-8##
##(α-β)^2=\dfrac{17}{4}##

For (c),
##α^3+β^3=(α+β)(α^2-αβ+β^2)##
##α^2+β^2=(α+β)^2-2αβ##
##α^2+β^2=(3.5)^2-4##
##α^2+β^2=\dfrac{33}{4}##
Therefore,
##α^3+β^3=(α+β)(α^2+β^2-αβ)##
##α^3+β^3=(-3.5)(8.25-2)##
##α^3+β^3=(-3.5)(6.25)##
##α^3+β^3=-\dfrac{175}{8}##

For part (d),
##α^3-β^3=\sqrt{\dfrac {17}{4}}⋅\dfrac {41}{4}=\dfrac {41}{4}⋅{\dfrac {\sqrt17}{2}}=\dfrac {41}{8}\sqrt 17##

For part (e),
We shall have,
##x^2-\dfrac{α^3+β^3}{αβ}x+αβ=0##
##x^2-\dfrac{175}{16}x+2=0##
##⇒16x^2-175x+32=0##

Bingo!

Last edited:
Parts b and c agree with the results I got, and pretty much follow the approach I used.

## 1. What is a quadratic equation?

A quadratic equation is a mathematical expression in the form of ax^2 + bx + c = 0, where a, b, and c are constants and x is a variable. It represents a parabola when graphed and has two solutions or roots.

## 2. What is the sum and product method for solving quadratic equations?

The sum and product method is a technique used to solve quadratic equations that involve both the sum and product of the roots. It involves finding two numbers that add up to the coefficient of the x-term and multiply to the constant term. These numbers are then used to factor the equation and find the roots.

## 3. When is the sum and product method useful?

The sum and product method is useful when the quadratic equation has both the sum and product of the roots given, or when the quadratic equation can be rewritten in the form of (x + a)(x + b) = 0, where a and b are the two numbers found using the sum and product method.

## 4. What are the steps to solve a quadratic equation using the sum and product method?

The steps to solve a quadratic equation using the sum and product method are:

1. Identify the coefficient of the x-term and the constant term.
2. Find two numbers that add up to the coefficient of the x-term and multiply to the constant term.
3. Write the equation in the form of (x + a)(x + b) = 0, where a and b are the two numbers found in the previous step.
4. Set each factor equal to 0 and solve for x.
5. The solutions or roots are the values of x found in the previous step.

## 5. Can the sum and product method be used for all quadratic equations?

No, the sum and product method can only be used for quadratic equations that involve both the sum and product of the roots. If the equation does not have both of these given, then the quadratic formula or other methods must be used to solve it.

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