Question

A quadratic equation, whose roots are \[\alpha \] and \[\beta \] can be written as \[\left( x-\alpha \right)\left( x-\beta \right)=0={{x}^{2}}-\left( \alpha +\beta \right)x+\alpha \beta \] i.e. \[a{{x}^{2}}+bx+c=a\left( x-\alpha \right)\left( x-\beta \right)\] .
A.True
B.False

Answer 1

Hint: In the given question, we have been asked that the given statement is true or false. The given statement is that a quadratic equation, whose roots are \[\alpha \] and \[\beta \] can be written as \[\left( x-\alpha \right)\left( x-\beta \right)=0={{x}^{2}}-\left( \alpha +\beta \right)x+\alpha \beta \] . In order to find out, we will need to solve \[\left( x-\alpha \right)\left( x-\beta \right)\] by using the distributive property of multiplication and simplifying this we will get the required result.
Formula used:
The distributive property of multiplication over addition or subtraction, i.e. when a number is multiplied by the sum or the difference of the numbers.
 \[\Rightarrow \left( a+b \right)\times \left( c+d \right)=\left( a\times c \right)+\left( a\times d \right)+\left( b\times c \right)+\left( b\times d \right)=ac+ad+bc+bd\]

Complete step-by-step answer:
We have given that,
A quadratic equation, whose roots are \[\alpha \] and \[\beta \] can be written as \[\left( x-\alpha \right)\left( x-\beta \right)\] .
Now,
We have,
 \[\left( x-\alpha \right)\left( x-\beta \right)\]
Using the distributive property of multiplication i.e. \[\left( a-b \right)\left( c-d \right)=ac-ad-bc+bd\]
Therefore,
 \[\Rightarrow \left( x-\alpha \right)\left( x-\beta \right)={{x}^{2}}-\beta x-\alpha x+\alpha \beta \]
Making the pair by taking out the common factor;
 \[\Rightarrow \left( x-\alpha \right)\left( x-\beta \right)={{x}^{2}}-\left( \alpha +\beta \right)x+\alpha \beta \]
Therefore,
A quadratic equation, whose roots are \[\alpha \] and \[\beta \] can be written as \[\left( x-\alpha \right)\left( x-\beta \right)=0={{x}^{2}}-\left( \alpha +\beta \right)x+\alpha \beta \] .
Hence, the given statement is true.
So, the correct answer is “Option A”.

Note: Here in the given question, students should note down that when we have given the roots of the quadratic equations. Let \[\alpha \] and \[\beta \] are the roots of the quadratic equation, then the formula for the quadratic equation is given by,
 \[\Rightarrow {{x}^{2}}-\left( \alpha +\beta \right)x+\alpha \beta \] , where \[\alpha +\beta \] is the sum of the roots and \[\alpha \beta \] is the products of the roots.
For solving these types of questions, we need to start with the last step, that is the roots can be written in the form \[\left( x-\alpha \right)\left( x-\beta \right)\] . Then we will need to reverse the last step to the first step and we will get the required equation.