Mathematics · Class 12 Science
Ch 3Quadratic Expressions — Class 12 Mathematics, concept-first.
Algebra lets us solve problems that would be difficult or impossible using arithmetic alone, and it underpins nearly every advanced branch of mathematics, science, engineering and industry that follows. It was not always written the way you see it today, though.
Key concepts
Hover a concept to preview it and jump to its most relevant Q&A.
Quadratic Expressions
A quadratic expression ax²+bx+c (a ≠ 0) is the algebraic backbone behind parabolas, and this chapter builds every tool needed to reason about it: the roots and the discriminant that classifies them, the neat sum/…
Most relevant Q&A
- For what values of $x$ the expression $x^2 - 5x - 14$ is positive?Preview
- Find the maximum value of the function $\dfrac{x^2 + 14x + 9}{x^2 + 2x + 3}$ over $\mathbb{R}$.Preview
- Find the values of m, for which the equation $x^2 - 15 - m(2x - 8) = 0$ have equal roots.Preview
- If $x$ is real, prove that $\dfrac{x}{x^2 - 5x + 9}$ lies between $-\dfrac{1}{11}$ and $1$.Preview
- Find the quadratic equation, the sum of whose roots is 7 and the sum of the squares of the roots is 25.Preview
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Introduction
Algebra lets us solve problems that would be difficult or impossible using arithmetic alone, and it underpins nearly every advanced branch of mathematics, science, engineering and industry that follow…
What Is a Quadratic Expression?
A quadratic expression in one variable is anything of the form
Solving the Quadratic Equation and the Discriminant
Every quadratic equation () can be solved by the same completing-the-square trick, and it always produces the two roots
Roots and Coefficients: Sum, Product, and Building an Equation from Given Roots
Once you have the root formula, adding and multiplying the two roots and produces two remarkably clean identities — the square-root parts cancel in the sum and simplify in the product:
Sign of a Quadratic Expression and How It Changes
For a fixed real quadratic ( real, ), a natural question is: for which real is positive, and for which is it negative? The discriminant answers this completely.
Maximum and Minimum Values
Completing the square, , does more than settle the sign question — it also pins down the single most extreme value ever takes, because a square is never negative.
Quadratic Inequations
A quadratic inequation in one variable is any statement of the form , , , or , with real and . Unlike a quadratic equation, which typically has just two solutions, a quadratic inequation usually has i…
Sample & Board Papers
Sample papers and previous-year board questions for this subject.
+−Show 13 questionsHide questions13 questions
- Q1For what values of $x$ the expression $x^2 - 5x - 14$ is positive?Preview
- Q2Find the maximum value of the function $\dfrac{x^2 + 14x + 9}{x^2 + 2x + 3}$ over $\mathbb{R}$.Preview
- Q3Find the values of m, for which the equation $x^2 - 15 - m(2x - 8) = 0$ have equal roots.Preview
- Q4If $x$ is real, prove that $\dfrac{x}{x^2 - 5x + 9}$ lies between $-\dfrac{1}{11}$ and $1$.Preview
- Q5Find the quadratic equation, the sum of whose roots is 7 and the sum of the squares of the roots is 25.Preview
- Q6Find the range of the expression $\dfrac{x^2 + x + 1}{x^2 - x + 1}$.Preview
- Q7Form the quadratic equation whose roots are $-3 \pm 5i$.Preview
- Q8Prove that $\dfrac{1}{3x+1} + \dfrac{1}{x+1} - \dfrac{1}{(3x+1)(x+1)}$ does not lie between $1$ and $4$, if $x$ is real.Preview
- Q9Determine the range of the expression $\dfrac{x+2}{2x^2+3x+6}$.Preview
- Q10Form quadratic equation whose roots are $7 \pm 2\sqrt{5}$.Preview
- Q11If $x$ is real, prove that $\frac{x}{x^2-5x+9}$ lies between $-\frac{1}{11}$ and $1$.Preview
- Q12Find the quadratic equation whose roots are $-3 \pm 5i$.Preview
- Q13Prove that $\dfrac{1}{3x + 1} + \dfrac{1}{x + 1} - \dfrac{1}{(3x + 1)(x + 1)}$ does not lie between $1$ and $4$, if $x$ is real.Preview