FINDING UNKNOWN INVOLVING SUM AND PRODUCT OF ROOTS WORKSHEET

Use the sum and product of roots formulas to answer the questions below :

1)  The roots of the equation x² – kx + k – 1 = 0 are α and 2α. Find the value(s) of k.

Solution

2)  The roots of the quadratic equation

x² + 6x + c are k and k - 1

Find the value of c.

Solution

3)  The roots of the quadratic equation

2x² - 9x + k

are m/2 and m - 3. Find the value of k.

Solution

4)  Find the values of m for which one root of the equation

4x² + 5 = mx

is three times the other root.

Solution

5)  One root of the equation 3x² - 4x + m = 0 is double the other. Find the roots, and the value of m.

Solution

6)  The roots of the equation

4x² - kx + 35 = 0

differ by one. Find the value of k.

Solution

Problem 1 :

Find k if the difference between the roots of the quadratic equation

x² – 4x + k = 0 is 2

Solution

Problem 2 :

Find the value of p such that the difference of the roots of the equation

x² – px + 8 = 0 is 2

Solution

Problem 3 :

Find the value of k such that the difference of the roots of the equation

2kx² – 20x + 21 = 0 is 2

Solution

Problem 4 :

Find k so that one root of the equation 2x² – 16x + k = 0 is twice the other. (Hint : One root = α, Other root = 2α)

Solution

Problem 5 :

Find k so that one root of the equation

k(x – 1)² = 5x – 7

is twice the other.                 Solution

Problem 6 :

Find k so that the roots of the quadratic equation

2x² + 3x + k = 0

are equal.                        Solution

Problem 7 :

If 1 – i and 1 + i are the roots of the equation

x² + ax + b = 0

where a, b ∈  r, then find the values of a and b.

Solution

Write the quadratic equation with Integral coefficients which have the following roots :

Problem 1 :

Roots : 2/5 and 4/3

Solution

Problem 2 :

Roots : 2/3 and 5/6

Solution

Problem 3 :

Roots : (3 + √5) and (3 - √5)

Solution

Problem 4 :

Roots : (2 + 3√2) and (2 - 3√2)

Solution

Problem 5 :

Roots : (3 + 4i) and (3 – 4i)

Solution

Problem 6 :

One root of 4 + √7

Solution

Problem 7 :

The equation x² + 2x + 5 = 0 has roots α and β. Use the roots method to find equation with integer coefficients which have the following roots.

a) 3α and 3β

b) α + 1 and β + 1

c) 1/α and 1/β

Solution

Problem 8 :

A ball is dropped from a window at a height of 81 feet. The function

h = -16x² + 81

represents the height (in feet) of the ball after x seconds. How long does it take for the ball to hit the ground?

Solution

Find the values of k for which the given quadratic equations have real and equal roots.

Problem 1 :

4x² + kx + 9 = 0

Solution

Problem 2 :

kx² - 5x + k = 0

Solution

Problem 3 :

x² - 7(3+k) + 4 - 2x(1 + k) = 0

Solution

Problem 4 :

Consider x² - 2x + m = 0. Find the discriminant, hence find the values of m for which the equation has

a) repeated roots

b) 2 distinct real roots

c)  no real roots

Solution

Problem 5 :

If -4 is a root of the quadratic equation x² + px - 4 = 0 and the quadratic equation 12x² + 4px + k = 0 has equal roots, the find the value of k.

Solution

Problem 6 :

If the equation (1+k²)x² + 2kqx + (q² - p²) = 0 has equal roots, then shown that q² = p²(1 + k²).

Solution