Quadratic Equations

• Quadratic equation: Introduction, Methods of solving quadratic equation, Nature of roots of quadratic equation, Relation between roots and coefficients, Applications of quadratic equations.
• When the degree of polynomial is 1 it is called a linear polynomial and if degree of a polynomial is 2 it is called a quadratic polynomial.
• The equation involving one variable and having 2 as the maximum index of the variable is called the quadratic equation.
• The values of a, b, c from equation ax² + bx + c = 0 are substituted in b²ac² and further simplified to obtain the roots of the equation.
• The formula used to solve quadratic equation is x = ± ±b²ac².
• If a = ±b², then a > b, if a = ±b², then a < b.
• To solve quadratic equations using formula, compare with ax² + bx + c = 0 and substitute the values of a, b, c.
• Out of the two roots, one can be represented by a and the other by b.
• The general form of quadratic equation is ax 2 + bx + c = 0, where a, b, c are real numbers and a ¹ 0.
• If the value of a polynomial p(x) is zero for x = a, then (x - a) is a factor of that polynomial.
• The equation 9 y 2 + 5 = 0 has as the only variable with a maximum index of 1, making it a quadratic equation.
• The equation 3 x 2 - 5 x + 3 = 0 has x as the only variable with a maximum index of 2, making it a quadratic equation.
• The equation (l + 2) (l - 5) = 0 has as the only variable with a maximum index of 2, making it a quadratic equation.
• The equation m3 - 5 m2 + 4 = 0 has as the only variable with a maximum index of 2, making it a quadratic equation.
• The solutions of x 2 = 2 x + 3, also known as the solutions of x 2 - 2 x - 3 = 0, are x = -1 or x = 3.
• The equation x 2 - 2 x - 3 = 0 can be solved graphically by comparing it with ax 2 + bx + c = 0, where a = 1, b = -2, c = -3.
• The graphs of equations y = x 2 and y = 2 x + 3 intersect at ( - 1, 1) and (3, 9), and the solutions of x 2 = 2 x + 3 are x = -1 or x = 3.
• The quadratic equation 5x - 3 = x2 is not a quadratic equation.
• Out of the following equations, find the equation having the sum of its roots - 5.
• The total number of trees in a column is x.
• The discriminant for the equation 2x2 - 5x + 2 = 0 is 2.
• The roots of x2 + kx + k = 0 are real and equal, find k.
• If 460 is divided by a natural number, quotient is 6 more than five times the divisor and remainder is 1.
• The area of a trapezium AB || CD is 33 cm2.
• The total number of trees in a row is y.
• The lengths of all sides of the trapezium ABCD can be found from the information given in the figure.
• The quadratic equation x2 - 15x + 8 = 0 has roots 3, 5.
• If a and b are the roots of the equation x2 - 13x + k = 0, then the difference between the roots is 7.
• If a and b are the roots of the equation x2 + 5x - 1 = 0, then find a3 + b3 and a2 + b2.
• The relation between roots of the quadratic equation and coefficients a and b are the roots of the equation ax2 + bx + c = 0 then, a + b = 0.
• The roots of the equation ax2 + bx + c = 0 are represented as a + b and a'.
• The roots of the equation x2 - 13x + k = 0 can be represented as a + b and a'.
• Ranjana wants to distribute 540 oranges among some students.
• Dinesh dug a square-shaped pond inside the farm to harvest rain water.
• If 30 students were more, each would get 3 oranges less.
• The length of Dinesh's agricultural farm at village Talvel is 10 meter more than twice the breadth.
• The square of a smaller number is twice the greater number.
• A tank fills completely in 2 hours if both the taps are open.
• If only one of the taps is open at the given time, the smaller tap takes 3 hours more than the larger one to fill the tank.
• The length and breadth of Dinesh's farm and of the pond are to be found.