(x-c)%2B(x-c)(x-a)%2B(x-a)(x-b)%3D0.jpeg "(x-b)(x-c)+(x-c)(x-a)+(x-a)(x-b)=0")
Factorization of a Quadratic Expression
In this article, we will explore the factorization of a quadratic expression, specifically the expression:
$(x-b)(x-c)+(x-c)(x-a)+(x-a)(x-b)=0$
This expression may seem complex at first glance, but by using the distributive property and combining like terms, we can simplify it and reveal its underlying structure.
Step 1: Expand the Expression
Let's start by expanding the expression using the distributive property:
$(x-b)(x-c)=x^2-xc-bx+bc$ $(x-c)(x-a)=x^2-ax-cx+ac$ $(x-a)(x-b)=x^2-ax-bx+ab$
Now, let's add the three expanded expressions:
$x^2-xc-bx+bc+x^2-ax-cx+ac+x^2-ax-bx+ab=0$
Step 2: Combine Like Terms
Next, we combine like terms:
$3x^2-(a+b+c)x+(ab+ac+bc)=0$
The Simplified Expression
The simplified expression reveals that it is a quadratic equation in the form of:
$ax^2+bx+c=0$
where $a=3$, $b=-(a+b+c)$, and $c=ab+ac+bc$.
Conclusion
In conclusion, we have successfully factorized the quadratic expression $(x-b)(x-c)+(x-c)(x-a)+(x-a)(x-b)=0$ and simplified it to reveal its underlying structure. This result can be used to solve various problems in mathematics, physics, and engineering, where quadratic equations play a crucial role.
Final Thoughts
The process of factorizing a quadratic expression can be tedious, but with the right approach, it can be simplified and made more manageable. By applying the distributive property and combining like terms, we can uncover the hidden structure of the expression and reveal its underlying beauty.
