Quadratic Equations
For what value of $k$ does the quadratic equation $x^2 - kx + 9 = 0$ have equal real roots?
What is the nature of the roots of the quadratic equation $2x^2 - 4x + 3 = 0$?
What is the standard form of a quadratic equation?
Define the discriminant of a quadratic equation and write its formula.
Write the quadratic formula (Sridharacharya formula) for solving a quadratic equation.
State the condition for a quadratic equation to have real and distinct roots.
State the condition for a quadratic equation to have no real roots.
Check whether x(2x + 3) = x² + 1 is a quadratic equation.
Find the discriminant of the quadratic equation 2x² - 4x + 3 = 0.
Determine whether x = 3 is a solution of the quadratic equation x² - 2x - 3 = 0.
Find the roots of the quadratic equation x² - 3x - 10 = 0 by factorization.
For what value of k does the quadratic equation kx(x - 2) + 6 = 0 have two equal roots?
Write the nature of roots of the quadratic equation 3x² - 4x + 5 = 0 without solving it.
Find the roots of the quadratic equation $2x^2 - 5x + 3 = 0$ by using the quadratic formula.
Explain the concept of Quadratic Equations in detail. Discuss their standard form, methods of finding their roots (factorization and quadratic formula), and the significance of the discriminant in determining the nature of roots.
A motor boat whose speed is 18 km/h in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream. Also, explain the theoretical concept of the discriminant of a quadratic equation and how it helps in determining the nature of roots.