Year 10 Quadratic Equations And Expressions And Inequalities

Question 1

Expand (x+1)^2

Answer 1

= (x + 1)(x + 1)
= x^2 + 2x + 1

Question 2

Completing the square

Answer 2

Form (x + a)² + b
Form a(x + b)² + c when the coefficient of x is not 1.

Question 3

x² + 8x in completed square form

Answer 3

(x + 4)² - 16
| |
Half of 8 |
4 is then squared, THIS VALUE MUST ALWAYS BE A NEGATIVE NUMBER.

Question 4

x² + 12x - 1 in completed square form

Answer 4

(x + 6)² - 36 - 1
| | |
Half of 12 | This value, -1, is added on after the expression.
Square of 6, THIS VALUE MUST ALWAYS BE A NEGATIVE NUMBER.
= (x + 6)² - 37

Question 5

2x ² + 8x + 7 in completed square form
(Coefficient of x is not 1)

Answer 5

1) factorise the first two terms: = 2(x ² + 4x) + 7
2) continue as normal: = 2((x + 2) ² - 4) + 7
= 2(x + 2)² - 8 + 7
= 2(x + 2) ² - 1

Question 6

Write 3 - 12x - x ² in form ax ² + bx + c

Answer 6

— x ² - 12x + 3

Question 7

Solve x ² + 4x - 7 = 0

Answer 7

1) Completed square form: (x + 2) ² - 4 - 7 = 0
(x + 2) ² - 11 = 0
2) Solve the equation: (x + 2) ² = 11
x + 2 = ± √11 You need a ± sign before √11 to avoid losing a solution.
x = - 2 ± √11

Question 8

Solve 3x ² - 30x + 71 = 0
(Co - efficient of x is not 1)

Answer 8

1) Completed square form
3(x ² - 10x) + 71 = 0
3((x - 5) ² - 25) + 71 = 0
3(x - 5) ² - 75 + 71 = 0
3(x - 5) ² - 4 = 0

2) Solve
3(x - 5) ² = 4
(x - 5) ² = 4/3
x - 5 = ±√4/3 ±√4/3 is used here to avoid losing a solution
x = 5 ±√4/3
x = 5 ± 2/√3
x = 5 ± 2√3 / 3

In a normal sum without this weird surf thing, so if x was 5+- root 4 instead
You would get 2 answers: x = 5+ root 4 = 7
OR
x = 5 - root 4 = 3

Question 9

Quadratic graph into equation
\ 2| /
1| /
—.—+—————
-1 -1| 1 2 3 4
-2x x
-3|_/

Answer 9

This is the first point NEGATIVE MEANS UNDER X AXIS in line with the x axis.
|
(x - 0)(x - 2)
|
This is the next point in line with the x axis. Negative because under x axis.

Question 10

What is the quadratic formula and when to use

Answer 10

. -b ± √b² - 4ac
x = ——————
. 2a
Example of use:
4x² + 3x + 1 = 0