
If one of the roots of the equation ${{x}^{2}}+ax-b=0$ is 1, then what is ( a – b) equal to?
( a ) – 1
( b ) 1
( c ) 2
( d ) - 2
Answer
160.8k+ views
Hint: In this question, we are given a quadratic equation with one of its roots and we have to find the value of (a – b). First, we compare the equation with the standard form of a quadratic equation and find out the value of a, b and c then we suppose the roots of the quadratic equation and after comparing the roots with the sum of roots and the product of roots, we find the value of ( a – b)
Formula Used: Sum of roots = $\dfrac{-b}{a}$
Product of roots = $\dfrac{c}{a}$
Complete step by step Solution:
Given quadratic equation is ${{x}^{2}}+ax-b=0$………………………. (1)
To find the roots of a quadratic equation we compare the equation (1) with the standard quadratic equation $a{{x}^{2}}+bx+c=0$, we get
a = 1 , b = a and c = -b
Let p and q be the roots of the given quadratic equation.
Then sum of roots ( p + q) = $-\dfrac{b}{a}=-\dfrac{a}{1}$ = - a
And the product of roots (pq) = $\dfrac{c}{a}=-\dfrac{b}{1}$= -b
We are given one of the roots of equation 1.
Let q = 1 then
p+q = -a
And p + 1 = -a
That is a = -p -1
Similarly pq = -b
That is p(1) = -b
Then b = -p
We have to find the value of a – b
That is a – b = ( -p-1) – (-p)
a – b = -p-1+p = - 1
Hence, the value of ( a – b ) = - 1
Therefore, the correct option is (a).
Note: Whenever we solve these types of questions where roots are given, we use the identity of the product of roots which is if x and y are the roots of any quadratic equation the value of xy will be equal to $\dfrac{ constant\, term}{coefficient\, of\, x^2}$ and sum of the roots that is x + y is equal to $\dfrac{ -coefficient\, of \,x}{coefficient\, of\, x^2}$ and by solving it we get the desired answer.
Formula Used: Sum of roots = $\dfrac{-b}{a}$
Product of roots = $\dfrac{c}{a}$
Complete step by step Solution:
Given quadratic equation is ${{x}^{2}}+ax-b=0$………………………. (1)
To find the roots of a quadratic equation we compare the equation (1) with the standard quadratic equation $a{{x}^{2}}+bx+c=0$, we get
a = 1 , b = a and c = -b
Let p and q be the roots of the given quadratic equation.
Then sum of roots ( p + q) = $-\dfrac{b}{a}=-\dfrac{a}{1}$ = - a
And the product of roots (pq) = $\dfrac{c}{a}=-\dfrac{b}{1}$= -b
We are given one of the roots of equation 1.
Let q = 1 then
p+q = -a
And p + 1 = -a
That is a = -p -1
Similarly pq = -b
That is p(1) = -b
Then b = -p
We have to find the value of a – b
That is a – b = ( -p-1) – (-p)
a – b = -p-1+p = - 1
Hence, the value of ( a – b ) = - 1
Therefore, the correct option is (a).
Note: Whenever we solve these types of questions where roots are given, we use the identity of the product of roots which is if x and y are the roots of any quadratic equation the value of xy will be equal to $\dfrac{ constant\, term}{coefficient\, of\, x^2}$ and sum of the roots that is x + y is equal to $\dfrac{ -coefficient\, of \,x}{coefficient\, of\, x^2}$ and by solving it we get the desired answer.
Recently Updated Pages
If there are 25 railway stations on a railway line class 11 maths JEE_Main

Minimum area of the circle which touches the parabolas class 11 maths JEE_Main

Which of the following is the empty set A x x is a class 11 maths JEE_Main

The number of ways of selecting two squares on chessboard class 11 maths JEE_Main

Find the points common to the hyperbola 25x2 9y2 2-class-11-maths-JEE_Main

A box contains 6 balls which may be all of different class 11 maths JEE_Main

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

Displacement-Time Graph and Velocity-Time Graph for JEE

Degree of Dissociation and Its Formula With Solved Example for JEE

Free Radical Substitution Mechanism of Alkanes for JEE Main 2025

Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

JEE Advanced 2025: Dates, Registration, Syllabus, Eligibility Criteria and More

JEE Advanced Weightage 2025 Chapter-Wise for Physics, Maths and Chemistry

NCERT Solutions for Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations

NCERT Solutions for Class 11 Maths In Hindi Chapter 1 Sets

NCERT Solutions for Class 11 Maths Chapter 6 Permutations and Combinations
