Mathematics
Introduction to Quadratic Equations
द्विघात समीकरणों का परिचय
Introduction to Quadratic Equations, part of the Class 10 Mathematics chapter Quadratic Equations, helps students recognise equations of degree two and write them in the standard form ax² + bx + c = 0, where a ≠ 0. Students learn the meaning of coefficients, variables, and constants, identify quadratic equations from examples, and understand how their roots or solutions relate to the equation. The topic builds a foundation for solving quadratic equations by methods such as factorisation and applying the quadratic formula.
Practice questions
What is the nature of the roots of the quadratic equation \(16x^2-24kx+9k^2=0\)?ExpertLevel 28If one root of (x^2+mx+36=0) is the reciprocal of the other, what can be said about (m)?ExpertLevel 28The roots of (x^2+px+25=0) are equal and positive. What is the value of (p)?ExpertLevel 28If (x=2) is not a root of (kx^2-7x+3=0), what condition must (k) satisfy?ExpertLevel 28Which quadratic equation has sum of roots (0) and product (-121)?ExpertLevel 28If in \(4x^2+kx+28=0\), the product of roots is (-2) times the sum of roots, what is (k)?ExpertLevel 28What is the sum of cubes of the roots of (x^2-5x-14=0)?ExpertLevel 28The length of a rectangle is \(x+3\) cm and its breadth is \(x-2\) cm. If its area is \(40\) cm², which is the correct quadratic equation in \(x\)?ExpertLevel 28If \(a=-2\), what type of statement does \((a+2)x^2+(a^2-4)x+7=0\) become?ExpertLevel 28What is the standard form of ((x-3)^2+(x+4)^2=(x-1)^2)?ExpertLevel 28If \(x=-1\) is a root of the quadratic equation \(x^2+(4k-1)x+3k=0\), what is the value of \(k\)?ExpertLevel 28The roots of the quadratic equation \(x^2-7x+3=0\) are \(\alpha\) and \(\beta\). What is the value of \(\alpha+\beta+3\alpha\beta\)?ExpertLevel 28If the roots of (x^2+px+q=0) are (4) and (p), what is the value of (p)?ExpertLevel 28If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2-12x+35=0\), what is the value of \((\alpha-5)(\beta-5)\)?ExpertLevel 28A right triangle has a base of \((x+3)\) units, a height of \((x+7)\) units, and an area of 55 square units. Which quadratic equation represents this situation?ExpertLevel 28If \(\alpha\) and \(\beta\) are the roots of \(x^2-13x+36=0\), what is the value of \((\alpha+\beta)^2\)?ExpertLevel 28The equation \\(x^2+10x+k=0\\) has no real roots. What is the correct condition on \\(k\\)?ExpertLevel 28If \(x^2-2ax+a^2-25=0\), what is the difference between the larger and smaller roots?ExpertLevel 28What is the standard form with integer coefficients of (\frac{(x+1)^2}{2}+\frac{(x-3)^2}{3}=10)?ExpertLevel 28If the roots of the equation \(x^2-2(k+1)x+k^2=0\) are real and distinct, what is the correct condition on \(k\)?ExpertLevel 28