USING DISCRIMINANT FIND THE MISSING COEFFICIENT

To find missing coefficient in a quadratic equation using the concept of discriminant, we have to be aware of nature of roots of a quadratic equation.

Without solving the equation completely, by using the formula b² - 4ac, we can easily figure out how the roots will be.

Problem 1 :

Find all the values of a such that

ax² + 3x + 5 = 0

has two real roots.

Solution :

Since the given quadratic equation has two real roots,

b² - 4ac > 0

By comparing the given equation with ax² + bx + c = 0, we get

a = a, b = 3 and c = 5

3² - 4(a)(5) > 0

9 - 20a > 0

-20a > -9

20a < 9

a < 9/20

Problem 2 :

Find all values of a such that ax² + 48x + 64 = 0 has one real root (a double root)

Solution :

Since the given quadratic equation has one real and equal roots.

b² - 4ac = 0

By comparing the given equation with ax² + bx + c = 0, we get

a = a, b = 48 and c = 64

48² - 4(a)(64) = 0

2304 - 256a = 0

-256a = -2304

a = 2304/256

a = 9

Problem 3 :

Find all the values of a such that 

ax² + 3x - 6 = 0

has two imaginary roots.

Solution :

Since the equation has two imaginary roots, we use the condition.

b² - 4ac < 0

a = a, b = 3 and c = -6

(-3)² - 4a(-6) < 0

9 + 24a < 0

24a < -9

a < -9/24

a < -3/8

Problem 4 :

Find all the values of such that 

2x² - 6x + c = 0

has two imaginary roots.

Solution :

Since it has two imaginary roots,

b² - 4ac < 0

a = 2, b = -6 and c = c

(-6)² - 4(2)c < 0

36 - 8a < 0

-8a < -36

a > 36/8

a > 9/2

Problem 5 :

Find all values of c such that -4x² + 8x + c = 0 has two real roots.

Solution :

Since it has two imaginary roots,

b² - 4ac > 0

a = -4, b = 8 and c = c

8² - 4(-4) c > 0

64 + 16c > 0

16c > -64

Dividing by -16 on both sides.

c > -64/16

c > -4