Quadratic and cubic equation -show that -(common roots)

In summary, to prove that a=b=c when the equations ax2+bx+c=0 and x3+3x2+3x+2=0 have two common solutions, we can factorize the cubic equation by dividing it by (x^2 + x +1). Then, we can use the formulas for solving quadratic and cubic equations to show that a=b=c.
  • #1
Sumedh
62
0

Homework Statement



If the equations ax2+bx+c=0 and x3+3x2+3x+2=0 have two common solutions then show a=b=c.

Homework Equations





The Attempt at a Solution



first equation will be the factor of second.
taking out common from first equation.


how to show a=b=c??
please provide hints.
 
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  • #2
Find a real root of the cubic equation. Write the cubic polynomial as the product of a quadratic factor and a linear factor.
 
  • #3
Sumedh said:

Homework Statement



If the equations ax2+bx+c=0 and x3+3x2+3x+2=0 have two common solutions then show a=b=c.

Homework Equations


The Attempt at a Solution



first equation will be the factor of second.
taking out common from first equation.how to show a=b=c??
please provide hints.

factorise the cubic.
 
  • #4
The easiest way to approach this is to divide (x^3 + 3x^2+ 3x + 2) by (x^2 + x +1) to prove that it's a factor. You don't actually need to factorize, just verify that the factor.
 
  • #5
can we solve by the following formulas

for quadratic eqs.
α+ß =-b/a

αß= c/a

for cubic equations
(Γ=gamma)

α+ß+Γ=-b/a

(αß)+(ßΓ)+(αΓ)=c/a

αßΓ=-d/a



i am trying to solve by this method but i hung up! in between?
 
  • #6
thank you very much i got it:smile:
 

FAQ: Quadratic and cubic equation -show that -(common roots)

What is a quadratic equation?

A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants and x is the variable. It has a highest degree of 2 and can have two possible solutions, called roots.

What is a cubic equation?

A cubic equation is a polynomial equation of the form ax^3 + bx^2 + cx + d = 0, where a, b, c, and d are constants and x is the variable. It has a highest degree of 3 and can have three possible solutions, called roots.

What does it mean to have common roots in quadratic and cubic equations?

Having common roots in quadratic and cubic equations means that both equations have the same solutions. In other words, the equations share one or more roots.

How can you show that two equations have common roots?

To show that two equations have common roots, you can use the discriminant. If the discriminant of both equations is equal to 0, then they have at least one common root. You can also solve both equations for x and check if the solutions are the same.

What is the significance of common roots in quadratic and cubic equations?

Having common roots in quadratic and cubic equations can provide insight into the relationship between the two equations. It can also be useful in solving systems of equations or finding the intersection points of graphs represented by the equations.

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