Quadratic Equat...

Only one of the roots of ax² + bx + c = 0, a ≠ 0, is zero, if

Find the value of .

For what values of c does the equation 2x² + 3x + c = 0 have imaginary roots?

If the equation 9x² + 6kx + 4 = 0, k > 0, has equal roots, then both the roots are equal to

The equation x - = 2 - has

If and are the roots of the quadratic equation ax² + bx + c = 0, then the value of is

It is known that y² - 11y + 4k = 0 has two distinct roots. It is also known that one factor is (y - k) and k is greater than zero. What is the difference between the sum and products of the roots?

For what value of m will the equation (m + 1)x² + 2(m + 3)x + m + 8 = 0 have equal roots?

The values of a for which the expression (a² - 1)x² + 2 (a - 1)x + 2 is positive for any x are:

If -4 is a root of the quadratic equation x² + px - 4 = 0 and the quadratic equation x² + px + k = 0 has equal roots, find the value of k.

If the equations ax² + bx + c = 0 and x² + 2x + 3 = 0 have common roots, then find the ratio a : b : c.

Solve for x:

If x = , then

^{_{If and are the roots of the equation ax}2 _{+ bx + c = 0, then the equation whose roots are is}}

For what values of k will the quadratic equation x² + 4kx + k² - k + 2 = 0 have equal roots?

Each side of a square measures 4 cm more than each side of another square. If the sum of their areas is 400 sq. cm, then what is the area of the larger square?

B's age is the square of the age of A. After 5 years, B will be 3 times as old as A. What is the difference between their ages?

A motor boat takes 12 hours to go upstream and come back again to the starting point. The total distance travelled during the whole journey is 70 km. The speed of motor boat in still water, if the current of a stream runs at 1 km/hr, is

Joy buys some packets of colours for Rs. 300. If the cost of each packet had been decreased by Rs. 5, then he would have bought 5 more packets. How much does each packet cost?

A boat can travel a distance of 8.5 km in one hour in still water. If the same boat travels upstream for 72 km and then returns to the starting point, taking 17 hours for the entire journey, then what is the speed of the stream?

How many positive values will 'x' take such that 4^{x + 1} - 18 × 2^x + 8 = 0?

Read the following statements carefully and choose the suitable.

Statement I: The quadratic equation ax² + bx + c = 0 has no real roots if b² - 4ac > 0.
Statement II: The number of possible values of m (m is a whole number) is 5 if the quadratic equation 4x² + 16x + 4m = 0 has real roots.

A rectangular park 60 m long and 40 m wide has two concrete crossroads running in the middle of the park and the rest of the park has been used as a lawn. What is the width of the road if the area of the lawn is 2109 m²?

State 'T' for true and 'F' for false for the following statements and choose the suitable option from the table given below them.

(i) If one of the roots of the quadratic equation 2x² - kx - 3 = 0 is -1, the other root is 5.
(ii) If α + β = 4 and α³ + β³ = 44, then α and β are the roots of 3x² - 12 x + 5 = 0.
(iii) If the roots of ax² + b = 0 are real and distinct, then ab > 0.
(iv) The quadratic equation (y - c)(y - d) + 2 = 0 has real roots if (c - d) ≥ (2)^3/2 and (c - d) ≤ -(2)^3/2.

In the given figure, ABC is a right triangle with B as the right angle. What is the length of AC?