   Question Answers

# If the zeros of the quadratic polynomial $\text{a}{{\text{x}}^{\text{2}}}\,\text{+}\,\text{bx}\,\text{+}\,\text{c,}\,\text{c}\,\ne \,\text{0}$ all equal, thenA) c and a have opposite signsB) c and b have opposite signsC) c and a have the same signsD) c and b have the same signs  Hint: As you can see in the question $\text{a}{{\text{x}}^{\text{2}}}\,\text{+}\,\text{bx}\,\text{+}\,\text{c,}\,\text{c}\,\ne \,\text{0}$ then it is clear that discriminant of the polynomial is zero using the equation for discriminant $\text{(i}\text{.e}\,{{\text{b}}^{\text{2}}}\,\text{-}\,\text{4ac)}$ we can further find which variables have same sign.

Complete step by step solution:
Now as given in the question that the zeros of polynomial $\text{a}{{\text{x}}^{\text{2}}}\,\text{+}\,\text{bx}\,\text{+}\,\text{c,}\,\text{c}\,\ne \,\text{0}$ are equal.
So the value of discriminant ‘D’ has to be zero.
$\therefore$ the equation for discriminant is,
${{\text{b}}^{\text{2}}}\,-\,\text{4ac}\,\text{=}\,\text{0}$
Now, shifting the term $4\text{ac}$
${{\text{b}}^{\text{2}}}\,=\,\text{4ac}$
We have to analyze one thing at this point that is the term $'{{\text{b}}^{\text{2}}}'$ on LHS can never be negative because the square of any number is always positive.
This means that the term on RHS i.e $4\text{ac}$ is also always positive.
Now, the term $'4\text{ac }\!\!'\!\!\text{ }$ to be always positive, it is necessary that both ‘a’ & ‘c’ should have the same sign (i.e either they both should be positive or both should be negative).

$\therefore$ Option (C) both c and a have the same sign is correct.

Note: A discriminant can be positive, zero or negative, and this determines how many solutions three are to a given quadratic equation. A positive discriminant indicates that quadratic has two distinct real roots. A discriminant zero indicates that the quadratic has a repeated real number solution.
CBSE Class 10 Maths Chapter 4 - Quadratic Equations Formula    What Happens if the Earth Stops Rotating?  The Idea of Time  What are the Functions of the Human Skeletal System?  The Ghat of the Only World Summary  The Perimeter of Rectangle Formula  Dark Side of the Moon  The Converse of Pythagoras Theorem  Measuring the Rate of Change of Motion  Important Questions for CBSE Class 10 Maths Chapter 4 - Quadratic Equations  Important Questions for CBSE Class 11 Maths Chapter 5 - Complex Numbers and Quadratic Equations  Important Questions for CBSE Class 7 English Honeycomb Chapter 10 - The Story of Cricket  Important Questions for CBSE Class 8 Science Chapter 10 - Reaching The Age of Adolescence  CBSE Class 8 Science Reaching The Age of Adolescence Worksheets  Important Questions for CBSE Class 11 English Snapshots Chapter 1 - The Summer of the Beautiful White Horse  Important Questions for CBSE Class 6 Social Science The Earth Our Habitat Chapter 3 - Motions of the Earth  Important Questions for CBSE Class 11 Chemistry Chapter 10 - The s-Block Elements  Important Questions for CBSE Class 6 English Honeysuckle Chapter 10 - The Banyan Tree  Important Questions for CBSE Class 6 Social Science The Earth Our Habitat Chapter 5 - Major Domains Of The Earth  CBSE Class 10 Maths Question Paper 2017  CBSE Class 10 Maths Question Paper 2020  Previous Year Question Paper of CBSE Class 10 English  Maths Question Paper for CBSE Class 10 - 2011  Maths Question Paper for CBSE Class 10 - 2008  Maths Question Paper for CBSE Class 10 - 2012  Maths Question Paper for CBSE Class 10 (2016)  Maths Question Paper for CBSE Class 10 - 2009  Maths Question Paper for CBSE Class 10 - 2010  Maths Question Paper for CBSE Class 10 - 2007  RS Aggarwal Class 10 Solutions - Quadratic Equations  NCERT Solutions for Class 10 Maths Chapter 4 Quadratic Equations  RD Sharma Solutions for Class 10 Maths Chapter 8 - Quadratic Equations  NCERT Solutions for Class 11 Maths Chapter 5  NCERT Exemplar for Class 10 Maths Chapter 4 - Quadratic Equations (Book Solutions)  NCERT Solutions for Class 10 Maths Chapter 4 Quadratic Equations in Hindi  RS Aggarwal Solutions Class 10 Chapter 10 - Quadratic Equations (Ex 10H) Exercise 10.8  RS Aggarwal Solutions Class 10 Chapter 10 - Quadratic Equations (Ex 10E) Exercise 10.5  RS Aggarwal Solutions Class 10 Chapter 10 - Quadratic Equations (Ex 10F) Exercise 10.6  NCERT Solutions for Class 10 Maths Chapter 4 Quadratic Equations (Ex 4.4) Exercise 4.4  