3x² + 2x - 7

Substitute x = -3 into the expression

3(-3)² + 2(-3) - 7

- x - = +

a² = a x a

(-3)² = - 3 x -3 = 9

3(-3)² + 2(-3) - 7 = 3 x -3 x -3 + 2 x -3 - 7

3(-3)² + 2(-3) - 7 = 3 x 9 - 6 - 7

3(-3)² + 2(-3) - 7 = 27 - 13

3(-3)² + 2(-3) - 7 = 14

(x - 2)(x + 4)

Take each express in the first bracket to multiply each expression in the second bracket and simplify the expressions.

(x - 2)(x + 4) = x(x + 4) - 2(x + 4)
(x - 2)(x + 4) = x x x + x x 4 - 2 x x - 2 x 4
(x - 2)(x + 4) = x² + 4x - 2x - 8
(x - 2)(x + 4) = x² + 2x - 8

a² = a x a

(x - 1)² = (x - 1) x (x - 1)

Expand the bracket by taking each expressions in the first bracket to multiply the second bracket and simplify.

(x - 1)² = x x (x - 1) - 1 x (x - 1)
(x - 1)² = x x x - 1 x x - 1 x x - 1 x -1
(x - 1)² = x² - x - x + 1

Note: - x - = +

(x - 1)² = x² -2x + 1

(x - 1)² - 1 = x² -2x + 1 - 1
(x - 1)² - 1 = x² -2x + 0
(x - 1)² - 1 = x² -2x

Note: (expression)² = The first number squared + 2 x the first number and the second number + second number squared.

(x - 1)² = x² + 2 x x x (-1) + (-1)²
(x - 1)² = x² - 2x + 1

Note: (-1)² = -1 x -1 = 1 (- x - = +)

a x a = a²

- x - = +

-a x -a = +a²

(2a - b)(a - b) = 2a(a - b) - b(a - b)

2a(a - b) - b(a - b) = 2a x a - 2a x b - b x a + b x b

2a(a - b) - b(a - b) = 2a² - 2ab - ab + b²

2a(a - b) - b(a - b) = 2a² - 3ab + b²

Difference in mathematics means subtraction.

Product means multiplication.

Sum means addition.

Hence difference between (2d)² and 2d² implies (2d)² - 2d²

NOTE: In (2d)² the square (2) is affecting both 2 and d since they are both in a bracket but in 2d², the square (2) only affects the d.

2d means 2 x d

2d² means 2 times d square, thus 2 x d²

d² = d x d

Hence 2d² = 2 x d x d

In (2d)², you can simplify the 2d and when done square the result, thus multiply the result by itself.

Since d = 3, substitute 3 into wherever you see d and simplify

2d = 2 x d, but d = 3

⇒ 2d = 2 x 3 = 6

(2d)² = (2 x 3)² = 6² = 6 x 6 = 36

2d² = 2 x 3² = 2 x 3 x 3 = 2 x 9 = 18

(2d)² - 2d² = 36 - 18 = 18

Find the L.C.M of 1 and 4 and use it to divide the entire expression to simplify

The L.C.M of 1 and 4 is 4, hence divide the entire expression by 4

NOTE: - x - = +, hence -1 x -3 = + 3

In -1(2y-3), the -1 multiplies everything in the bracket

Principle: a(b+c) = a x b + a x c

a(b-c) = a x b - a x c

In 12y - 2y = 10y because they are like terms and can be simplified further

Expand simply means multiply and simplify

Principle

(a+b)(c+d) = a(c+d)+b(c+d)

(a-b)(c+d) = a(c+d)-b(c+d)

(7r-5)(3r+4) = 7r(3r+4)-5(3r+4)

7r(3r+4)-5(3r+4) = 7rx3r + 7rx 4 - 5x3r-5x4

7rx3r + 7rx 4 - 5x3r-5x4 = 21r² + 28r-15r-20

21r² + 28r-15r-20 = 21r² + 13r-20

NOTE: r x r = r²

a = a¹

From the law of indices, a^m x aⁿ = a^{m + n}

r x r = r¹ x r¹ = r¹⁺¹ = r²

x = 4, y = 7

2xy + 3(x+y) = 2 x 4 x 7 + 3(4+7)

2xy + 3(x+y) = 56 + 3(11)

2xy + 3(x+y) = 56 + 3 x 11

2xy + 3(x+y) = 56 + 33 = 89

Expand the brackets first

Concept

a(b + c) = a x b + a x c

a(b - c) = a x b - a x c

3(5a² + 2c) = 3 x 5a² + 3 x 2c

3(5a² + 2c) = 15a² + 6c

- 2a(1 - 3a) = - 2a x 1 - 2a x - 3a

NOTE: - x - = + and a x a = a²

- 2a(1 - 3a) = - 2a + 6a²

3(5a² + 2c) - 2a(1 - 3a) - 6c = 15a² + 6c - 2a + 6a² - 6c

Group like terms

3(5a² + 2c) - 2a(1 - 3a) - 6c = 15a² + 6a² - 2a + 6c - 6c

3(5a² + 2c) - 2a(1 - 3a) - 6c = 21a² - 2a + 0

3(5a² + 2c) - 2a(1 - 3a) - 6c = 21a² - 2a

Subtract (7x-3) from (5-3x) ⇒ (5-3x) - (7x-3)

(5-3x) - (7x-3) = (5-3x) - 1(7x-3)

NOTE: Every number is multiplied by 1 which is the same number or has no effect, thus - (7x-3) = - 1(7x-3)

Applying the distributive property, a(x+y) = axx+axy

⇒ - 1(7x-3) = -1x7x-1x(-3) = -7x+3

NOTE: - x - = +

(5-3x) - 1(7x-3) = 5-3x -7x+3

Grouping like terms ⇒ 5-3x -7x+3 = 5+3 -3x-7x

⇒ 5-3x -7x+3 = 8 -10x

Concept

(a+b)(c+d) = a x (c+d) + b x (c+d)

a(b+c) = a x (b+c) = a x b + a x c

⇒ (a - 2)(2a+3) = a x (2a+3) - 2 x (2a+3)

a x (2a+3) = a x 2a + a x 3 = 2a² + 3a

NOTE: a x a = a¹ x a¹ = a¹⁺¹ = a²

-2(2a+3) = -2 x 2a + -2 x 3 = -4a - 6

⇒ (a - 2)(2a+3) = 2a²+3a - 4a - 6

⇒ (a - 2)(2a+3) = 2a²- a - 6

NOTE: 3a - 4a = -1a = - a

Change the ratio to fraction form:

$\frac{\mathrm{x\; +\; 2}}{\mathrm{x\; -\; 2}}$ = $\frac{1}{2}$

Cross multiply:

2(x + 2) = 1(x - 2)

2x + 4 = x - 2

2x - x = -2 - 4

x = -6

Note: in an examination, always check your answer by substituting it back into the original equation if you have enough time.

Verification

Substituting x = -6 into the original ratio:

(-6 + 2) : (-6 - 2) = -4 : -8 = 1 : 2

Notes:

1. The negatives cancel out
2. 4 divides itself 1 and 8, 2 times

∴ The answer is correct.

p²(q - 1) = p²(q - 1)

p = 2 and q = $\frac{3}{4}$

Substitute the values:

p²(q - 1) = (2)²($\frac{3}{4}$ - 1)

2² = 2 x 2 = 4

Every whole number is being divided by 1 but not written as a fraction.

1 = $\frac{1}{1}$

p²(q - 1) = 4($\frac{3}{4}$ - $\frac{1}{1}$)

p²(q - 1) = 4($\frac{\mathrm{3\; x\; 1\; -\; 1\; x\; 4}}{4}$)

p²(q - 1) = 4($\frac{\mathrm{3\; -\; 4}}{4}$)

p²(q - 1) = 4($\frac{-1}{4}$)

p²(q - 1) = 4 x $\frac{-1}{4}$

p²(q - 1) = $\frac{\mathrm{4\; x\; -1}}{4}$

p²(q - 1) = $\frac{-4}{4}$

p²(q - 1) = -1

$\frac{3}{4}$x = 2 + $\frac{1}{4}$

Get rid of the fractions by multiplying both sides by 4.

4 x $\frac{3}{4}$x = 4 x 2 + 4 x $\frac{1}{4}$

3x = 8 + 1

3x = 9

Divide both sides by 3

x = $\frac{9}{3}$

x = 3

x - 5(3 - 2x) - 12x + 7 = x - 5(3 - 2x) - 12x + 7

Expand the brackets:

x - 15 + 10x - 12x + 7 = x - 15 + 10x - 12x + 7

Note: - x - = +. -5 x -2x = +10x

Group like terms:

x - 15 + 10x - 12x + 7 = x + 10x - 12x - 15 + 7

Simplify:

x - 15 + 10x - 12x + 7 = -x - 8