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Quadratic Formula and Number of Solutions

In this worksheet, students select the quadratic equation that can be solved using the given quadratic formula.

'Quadratic Formula and Number of Solutions' worksheet

Key stage:  KS 4

Curriculum topic:  Algebra

Curriculum subtopic:  Solve Quadratic Equations by Factorising

Difficulty level:  

down

Worksheet Overview

QUESTION 1 of 10

A quadratic equation can be solved using the quadratic formula, provided

that a solution exists. 

 

The quadratic equation

ax2 + bx + c = 0

 

where a ≠ 0, has up to two solutions given by

 

 

The part under the square root, b2 - 4ac, is called the discriminant and must be positive or zero for it to be "square rootable".

So....

In some cases, there will be no real solution and this occurs when the expression inside the square root, the discriminant, is negative. 

i.e. when b2 - 4ac < 0

 

In some cases, there will be only one real solution and this occurs when the expression inside the square root, the discriminant, is zero. 

i.e. when b2 - 4ac = 0

 

In all other cases, there will be two real solutions and this occurs when the expression inside the square root, the discriminant, is positive. 

i.e. when b2 - 4ac > 0

 

Example

State the number of solutions to:

3x2 - 7x + 5 = 0

 

Answer

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

b2 = 49

4ac = 4 x 3 x 5 = 60

So the discriminant,  b2 - 4ac < 0

i.e. No real solutions.

State the number of solutions to:

 

2x2 + 7x + 5 = 0

2 real solutions

1 real solution

No real solutions

State the number of solutions to:

 

4x2 + 7x + 5 = 0

2 real solutions

1 real solution

No real solutions

State the number of solutions to:

 

4x2 - 3x + 2 = 0

2 real solutions

1 real solution

No real solutions

State the number of solutions to:

 

4x2 - 12x + 9 = 0

2 real solutions

1 real solution

No real solutions

State the number of solutions to:

 

3x2 + 9x + 2 = 0

2 real solutions

1 real solution

No real solutions

State the number of solutions to:

 

3x2 - 5x + 2 = 0

2 real solutions

1 real solution

No real solutions

State the number of solutions to:

 

3x2 + 2x - 5 = 0

2 real solutions

1 real solution

No real solutions

State the number of solutions to:

 

x2 - 10x + 25 = 0

2 real solutions

1 real solution

No real solutions

State the number of solutions to:

 

-2x2 - 5x + 2 = 0

2 real solutions

1 real solution

No real solutions

State the number of solutions to:

 

-2x2 + 5x - 7 = 0

2 real solutions

1 real solution

No real solutions

  • Question 1

State the number of solutions to:

 

2x2 + 7x + 5 = 0

CORRECT ANSWER
2 real solutions
EDDIE SAYS
a = 2, b = 7, c = 5
b² - 4ac = 7² - 4 × 2 × 5 = 49 - 40 > 0
  • Question 2

State the number of solutions to:

 

4x2 + 7x + 5 = 0

CORRECT ANSWER
No real solutions
EDDIE SAYS
a = 4, b = 7, c = 5
b² - 4ac = 7² - 4 × 4 × 5 = 49 - 80 < 0
  • Question 3

State the number of solutions to:

 

4x2 - 3x + 2 = 0

CORRECT ANSWER
No real solutions
EDDIE SAYS
a = 4, b = -3, c = 2
b² - 4ac = (-3)² - 4 × 4 × 2 = 9 - 32 < 0
  • Question 4

State the number of solutions to:

 

4x2 - 12x + 9 = 0

CORRECT ANSWER
1 real solution
EDDIE SAYS
a = 4, b = -12, c = 9
b² - 4ac = (-12)² - 4 × 4 × 9 = 144 - 144 = 0
  • Question 5

State the number of solutions to:

 

3x2 + 9x + 2 = 0

CORRECT ANSWER
2 real solutions
EDDIE SAYS
a = 3, b = 9, c = 2
b² - 4ac = 9² - 4 × 3 × 2 = 81 - 24 > 0
  • Question 6

State the number of solutions to:

 

3x2 - 5x + 2 = 0

CORRECT ANSWER
2 real solutions
EDDIE SAYS
a = 3, b = -5, c = 2
b² - 4ac = (-5)² - 4 × 3 × 2 = 25 - 24 > 0
  • Question 7

State the number of solutions to:

 

3x2 + 2x - 5 = 0

CORRECT ANSWER
2 real solutions
EDDIE SAYS
a = 3, b = 2, c = -5
b² - 4ac = 2² - 4 × 3 × -5 = 4 - -60 > 0
  • Question 8

State the number of solutions to:

 

x2 - 10x + 25 = 0

CORRECT ANSWER
1 real solution
EDDIE SAYS
a = 1, b = -10, c = 25
b² - 4ac = (-10)² - 4 × 1 × 25 = 100 - 100 = 0
  • Question 9

State the number of solutions to:

 

-2x2 - 5x + 2 = 0

CORRECT ANSWER
2 real solutions
EDDIE SAYS
a = -2, b = -5, c = 2
b² - 4ac = (-5)² - 4 × -2 × 2 = 25 - -16 > 0
  • Question 10

State the number of solutions to:

 

-2x2 + 5x - 7 = 0

CORRECT ANSWER
No real solutions
EDDIE SAYS
a = -2, b = 5, c = -7
b² - 4ac = 5² - 4 × -2 × -7 = 25 - 56 < 0
---- OR ----

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