Question

Find the value of the discriminant for all the equations. 1.5q2 +2q–3=0 2.3u2 -4u+1=0 3.7s2 +9s–4=0 4.6w2 –8w+5=0 5.5p2 +7p–3=0 6.9f2 -5f+2=0 7.2n2 +6n–4=0 8.3d2 -3d+4=0 9.9v2 +4v–7=0 10.4c2 +8c–5=0

Answer 1

The question is:

Find the value of the discriminant for all the following equations:

1. 5q² + 2q - 3 = 0

2. 3u² - 4u + 1 = 0

3. 7s² + 9s - 4 = 0

4. 6w² - 8w + 5 = 0

5. 5p² + 7p - 3 = 0

6. 9f² - 5f + 2 = 0

7. 2n² + 6n - 4 = 0

8. 3d² - 3d + 4 = 0

9. 9v² + 4v - 7 = 0

10. 4c² + 8c - 5 = 0

Answer:

1. 5q² + 2q - 3 = 0

Discriminant is 64

2. 3u² - 4u + 1 = 0

Discriminant is 4

3. 7s² + 9s - 4 = 0

Discriminant is 193

4. 6w² - 8w + 5 = 0

Discriminant is -56

5. 5p² + 7p - 3 = 0

Discriminant is 109

6. 9f² - 5f + 2 = 0

Discriminant is -47

7. 2n² + 6n - 4 = 0

Discriminant is 68

8. 3d² - 3d + 4 = 0

Discriminant is -39

9. 9v² + 4v - 7 = 0

Discriminant is 268

10. 4c² + 8c - 5 = 0

Discriminant is 144.

Explanation:

The quadratic equation

ax² + bx + c = 0

can be solved using the quadratic formula

x = [-b ± √(b² - 4ac)]/2a

Here,

b² - 4ac

is called the discriminant

Let D = b² - 4ac

We want to find the discriminant of the following:

1. 5q² + 2q - 3 = 0

a = 5, b = 2, c = -3

D = 2² - 4(5)(-3)

= 4 + 60

= 64

Discriminant is 64

2. 3u² - 4u + 1 = 0

a = 3, b = -4, c = 1

D = (-4)² - 4(3)(1)

= 16 - 12

= 4

Discriminant is 4

3. 7s² + 9s - 4 = 0

a = 7, b = 9, c = -4

D = 9² - 4(7)(-4)

= 81 + 112

= 193

Discriminant is 193

4. 6w² - 8w + 5 = 0

a = 6, b = -8, c = 5

D = (-8)² - 4(6)(5)

= 64 - 120

= -56

Discriminant is -56

5. 5p² + 7p - 3 = 0

a = 5, b = 7, c = -3

D = 7² - 4(5)(-3)

= 49 + 60

= 109

Discriminant is 109

6. 9f² - 5f + 2 = 0

a = 9, b = -5, c = 2

D = (-5)² - 4(9)(2)

= 25 - 72

= -47

Discriminant is -47

7. 2n² + 6n - 4 = 0

a = 2, b = 6, c = -4

D = 6² - 4(2)(-4)

= 36 + 32

= 68

Discriminant is 68

8. 3d² - 3d + 4 = 0

a = 3, b = -3, c = 4

D = (-3)² - 4(3)(4)

= 9 - 48

= -39

Discriminant is -39

9. 9v² + 4v - 7 = 0

a = 9, b = 4, c = -7

D = 4² - 4(9)(-7)

= 16 + 252

268

Discriminant is 268

10. 4c² + 8c - 5 = 0

a = 4, b = 8, c = -5

D = 8² - 4(4)(-5)

= 64 + 80

= 144

Discriminant is 144.

Answer 2

Case 1: a = 1.5, b = 2, c = -3

`D = 22`

Case 2: a = 2.3, b = -4, c = 1

`D = 6.8`

Case 3: a = 3.7, b = 9, c = -4

`D = 140.2`

Case 4: a = 4.6, b = -8, c = 5

`D = -28`

Case 5: a = 5.5, b = 7, c = -3

`D = 115`

Case 6: a = 6.9, b = -5, c = 2

`D = -30.2`

Case 7: a = 7.2, b = 6, c = -4

`D = 151.2`

Case 8: a = 8.3, b = -3, c = 4

`D = -123.8`

Case 9: a = 9.9, b = 4, c = -7

`D = 293.2`

Case 10: a = 10.4, b = 8, c = -5

`D = 272`

Explanation:

For all second order polynomial, the discriminant is equal to:

`D = b^(2)-4\cdot a \cdot c`

Case 1: a = 1.5, b = 2, c = -3

`D = 22`

Case 2: a = 2.3, b = -4, c = 1

`D = 6.8`

Case 3: a = 3.7, b = 9, c = -4

`D = 140.2`

Case 4: a = 4.6, b = -8, c = 5

`D = -28`

Case 5: a = 5.5, b = 7, c = -3

`D = 115`

Case 6: a = 6.9, b = -5, c = 2

`D = -30.2`

Case 7: a = 7.2, b = 6, c = -4

`D = 151.2`

Case 8: a = 8.3, b = -3, c = 4

`D = -123.8`

Case 9: a = 9.9, b = 4, c = -7

`D = 293.2`

Case 10: a = 10.4, b = 8, c = -5

`D = 272`