# Solve the quadratic equation involving sum and product

• chwala
In summary, for part (i), we can use the given equations to find the value of α+β and αβ, and then using the formula for the sum of cubes, we can find the value of α^3+β^3. For part (ii), we can use the given formula to simplify the expression and then solve for x to get a quadratic equation.
chwala
Gold Member
Homework Statement
See attached
Relevant Equations
sum/product

For part (i),
##(x-α)(x-β)=x^2-(α+β)x+αβ##
##α+β = p## and ##αβ=-c##
therefore,##α^3+β^3=(α+β)^3-3αβ(α+β)##
=##p^3+3cp##
=##p(p^2+3c)##

For part (ii),
We know that; ##tan^{-1} x+tan^{-1} y##=##tan^{-1}\left[\dfrac {tan^{-1} x+tan^{-1} y}{1-tan^{-1} x⋅tan^{-1} y}\right]## then it follows that,
##tan^{-1}\left[ \frac {x}{c}\right]+tan^{-1} x##=##tan^{-1}\left[\dfrac {\dfrac{x}{c} + x}{1- \frac{x}{c}⋅ x}\right]##
We now have;
##tan^{-1}\left[\dfrac {\dfrac{x}{c} + x}{1- \frac{x}{c}⋅ x}\right]##=## tan^{-1}c##
##\left[\dfrac {\dfrac{x}{c} + x}{1- \frac{x}{c}⋅ x}\right]##=##c##
##\left[\dfrac {\dfrac{x}{c} + x}{1- \frac{x^2}{c}}\right]##=##c##
##\dfrac {x}{c}##+##x##=##c####(1##-##\dfrac{x^2}{c})##...from this we get,
##x^2+(\dfrac {1}{c}+1)x-c=0##

We know that, ##α+β = p## and ##αβ=-c##
it follows that
##-(β+α)##=##\frac {1}{c}+1## and ##αβ=-c##
then using,
##-(β+α)##=##\dfrac {1}{c}+1##
##-(β+α)##=##\dfrac {1+c}{c}##
##-p##=##\dfrac {1+c}{c}##
##-pc=1+c##
##⇒pc+c+1=0## Bingo,
I would appreciate any feedback on my steps...as i do not have markscheme or rather the solutions. Cheers guys.

Last edited:
You didn't do the last part of (1) where you have to find a quadratic equation?

For, the last part of (1), we shall have the factors;
##x=α^3## and ##x=β^3##, thus our quadratic equation will be of the form,
##(x-α^3)(x-β^3)##
##=x^2-(α^3+β^3)x+α^3β^3##
##=x^2-(p^3+3pc)x-c^3##

## What is a quadratic equation?

A quadratic equation is an algebraic equation in the form of ax^2 + bx + c = 0, where a, b, and c are constants and x is the variable. It represents a parabola when graphed and can have two solutions.

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

The sum and product method is a technique for solving quadratic equations by finding two numbers that add up to the coefficient of x and multiply to the constant term. These numbers can then be used to factor the equation into two binomials.

## 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 have a coefficient of 1 for x^2 and a constant term. If these conditions are not met, a different method, such as the quadratic formula, must be used.

## What are the steps for solving a quadratic equation using the sum and product method?

The steps for solving a quadratic equation using the sum and product method are as follows:

1. Identify the coefficients a, b, and c in the equation ax^2 + bx + c = 0.
2. Find two numbers that add up to b and multiply to c.
3. Write the equation in the form of (x + m)(x + n) = 0, where m and n are the two numbers found in the previous step.
4. Set each binomial equal to 0 and solve for x.
5. The solutions to the equation are the values of x that make each binomial equal to 0.

## What are some real-world applications of solving quadratic equations using the sum and product method?

The sum and product method can be used to solve problems involving maximizing profits, finding dimensions of geometric shapes, and predicting the trajectory of projectiles. It can also be used in engineering and physics to solve problems related to motion and forces.

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