2.02 Algebraic Manipulation

Substitution

Replacing variables with their values (these values may be numbers or another algebraic expression) to either solve equations, find other values in formulae, or compute the values of expressions.

How do you evaluate expressions by substitution?

1. Write the original expression.
2. Replace the variables with the given numerical values.


1. Remember to use brackets around negative numbers.
3. Simplify/compute any calculations where possible.

Example: Calculate 18x + 42 where x = 45.

18x + 42 =

18(45) + 42 =

810 + 42 =

852

Like terms

Terms containing the same variables raised to the same powers, coefficients can be different. Example: xy and 3yx are like terms (multiplication can be in any order), 3x^2 and 42x^2 are also examples of like terms.

How can you collect like terms?

Like terms are usually separated by + (addition sign) or - (subtraction sign). The action of the sign is performed with the like term after it on the like term before it. E.g: 3x + 5x, the addition belongs to 5x (we are adding with the 5x), and this action is performed to 3x → 3x + 5x is the same as adding 5x to 3x (though addition is commutative so 5x + 3x would yield the same result), which gives us 8x.


1. If the like terms are separated by an addition sign (+) then add the coefficients of the like terms and multiply the result of this sum to the like variable to get the final answer.


1. 3xy + 18xy = (3 + 18) \* xy = 21xy
2. If the like terms are separated by a subtraction sign (-) then subtract the coefficient of the like term after the subtraction sign from the coefficient of the like term before the subtraction sign and multiply the result of this subtraction to the like variable to get the final answer.


1. 18xy - 3xy = (18 - 3) \* xy = 15xy

How do you know whether the first term is positive or negative?

No arithmetic sign in front of first term → positive

\- sign in front of first term → negative

How do you multiply a single term over a bracket?

Term outside the bracket is the factor.


1. Multiply the factor with each term inside the bracket.
2. Remember to keep the - sign when multiplying negative numbers, it’s helpful to put brackets around negative terms.


1. Example: 3(18x - 42) = 3(18x) - 3(42) = 54x - 132
3. If there are multiple bracketed expressions added or subtracted together, then expand each bracket expression separately and subsequently collect the like terms.


1. Example: Evaluate 3(18x - 42) - 9(9x + 45).

3(18x - 42) - 9(9x + 45) =

3(18x) - 3(42) - 9(9x) - 9(45) =

54x - 132 - 81x - 405 =

\-27x - 537

How do you expand 2 bracketed expressions multiplied together?

Every term in the first bracket must be multiplied to every term in the second term, then collect the like terms. It’s useful to put negative numbers in brackets. Example:

(xy + 18x + 42)(xy + 9y - 18) =

xy(xy) + xy(9y) - xy(18) + 18x(xy) + 18x(9y) + 18x(-18) + 42(xy) + 42(9y) + 42(-18) =

x^2 \* y^2 + 9xy^2 - 18xy + 18x^2 \* y + 162xy - 324x + 42xy + 378y - 756 =

x^2 \* y^2 + 18x^2 \* y + 186xy - 324x + 378y + 9xy^2 - 756

If each of the 2 brackets have 2 terms, then use the FOIL acronym to expand the brackets:


1. First: Multiply the first term of the first bracket to the first term of the second bracket. (Both are the first terms).
2. Outer: Multiply the first term of the first bracket to the second term of the second bracket (visually these are the “outer” terms).
3. Inner: Multiply the second term of the first bracket to the first term of the second bracket (visually these are the “inner” terms).
4. Last: Multiply the second term of the first bracket to the second term of the second bracket. (Both are the last terms).
5. Then collect the like terms.

Example:

(x + 18) (x + 45) =

x(x) + x(45) + 18(x) + 18(45) =

x^2 + 45x + 18x + 810 =

x^2 + 63x + 810

How do you expand 3 or more bracketed expressions multiplied together?

1. Multiply the first 2 bracketed expressions as usual or using the FOIL method like previously explained.
2. Simplify the product obtained from step 1 by collecting like terms.
3. Multiply the longer bracket expression obtained from step 2 by the third bracket.
4. If there are more than 3 bracketed expressions multiplied together, then simplify the product obtained from step 3 by collecting like terms.
5. Multiply the longer bracket expression obtained from step 4 by the fourth bracket.
6. Repeat this until all bracketed expressions have been multiplied together.

Factorization

Opposite of expansion, it’s the process of writing an expression or term as the product of all its factors.

For example: factorizing 3x + 9 will give you 3(x + 3).

How do fully factorize an expression?

1. Find the HCF of all terms in the expression.
2. Divide each term by the HCF.
3. Write each term as HCF \* (term/HCF).
4. Add or subtract the terms from step 3 to each other to get the final expression.
5. Write the HCF outside the brackets, then write the remaining expression inside the brackets.
6. If there are any more common factors amongst terms inside the remaining bracketed expression, then factorize the remaining bracketed expression fully until there are no common factors, and multiply this factorization result to the HCF.

Example: Fully factorize 4x^5 + 20x^4 + 100x^2

HCF(4x^5, 20x^4, 100x^2) = 4x^2

4x^5 / 4x^2 = x^3 → 4x^5 = 4x^2 \* x^3

20x^4 / 4x^2 = 5x^2 → 20x^4 = 4x^2 \* 5x^2

100x^2 / 4x^2 = 25 → 100x^2 = 4x^2 \* 25

4x^2 \* x^3 + 4x^2 \* 5x^2 + 4x^2 \* 25 =

4x^2 (x^3 + 5x^2 + 25)

How do you factorize by grouping?

If the common factor is an expression in brackets:


1. The common bracket can be “taken out” like a common factor, then multiplied to another bracket containing the remaining expression.


1. Example: Fully factorize 3x(t + 45) - 42(t + 45)
2. 3x(t + 45) - 42(t + 45) =
3. (t + 45) (3x - 42) =
4. (t + 45) 3 (x - 14) =
5. 3 (t + 45) (x - 14)


2. Sometimes the common bracket may not be written out, so you have to group terms with common factors together to find the common bracket.


1. Example: Fully factorize xy + qy + px + pq
2. xy + qy + px + pq =
3. yx + yq + px + pq =
4. y(x + q) + p(x + q) =
5. (x + q) (y + p)

How do you check if an expression is factorized correctly?

Expand the factorized expression/brackets and check whether the expanded expression is the same as the original expression.

Not same → incorrect expansion

Same → correct expansion

Quadratic expression

Expressions with 1 variable whose highest power it’s raised to is 2. Specifically, it is an expression in the form of:

ax^2 + bx + c, where a, b, c, are all constants and a ≠ 0. For example: 3x^2 + 45x - 9 is a quadratic expression.

Monic quadratic expression

Quadratic expression where the coefficient of x^2 (i.e. a) is equal to 1.

How to factorize monic quadratic expressions?

x^2 + bx + c = (x + m) (x + n) = x^2 + nx + mx + mn = x^2 + (m + n) x + mn

We need to find 2 numbers m and n such that:

m + n = b

mn = c

You can factorize the monic quadratic expression by inspection (fastest for monic quadratic expressions) or grouping (slower but more reliable and can be used for non-monic quadratic expressions).

Inspection (trial and error) example:

x^2 - 2x - 8.

2 numbers that multiply to -8 and add to -2 are -4 and 2:

(x - 4) (x + 2) = x^2 + 2x - 4x - 8 = x^2 - 2x - 8.

This works because its expansion gives the correct original quadratic expression, so the factorized form of x^2 - 2x - 8 is (x - 4) (x + 2)

Grouping example:

x^2 - 2x - 8.

2 numbers that multiply to -8 and add to -2 are -4 and 2:

x^2 -4x + 2x - 8 =

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

(x + 2) (x - 4)

Non-monic quadratic expression

Quadratic expression where the coefficient of x^2 (i.e. a) is not equal to 1.

How to factorize non-monic quadratic expressions?

ax^2 + bx + c = (px + q) (rx + s) = prx^2 + psx + qrx + qs = prx^2 + (ps + qr) x + qs.

We need to find 2 numbers ps and qr such that:

ps + qr = b.

pqrs = pr(qs) = ac

We can then combine these 2 terms (ps and qr) with prx^2 and qs to factorize by grouping.

Example: Factorize 2x^2 - 5x - 3.

We must find 2 numbers ps and qr such that:

ps + qr = b, which is -5 in this case.

pqrs = ac, which is -6 in this case.

ps = -6, qr = 1

2x^2 - 6x + x - 3 =

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

(2x + 1) (x - 3)

Factorizing difference of squares

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

(a - b) (a + b) = a^2 + ab - ba - b^2 = a^2 + ab - ab - b^2 = a^2 - b^2

General methods for quadratics factorization

If discriminant (b^2 - 4ac) of the quadratic formula is a perfect square, then the quadratic expression can be factorized (i.e. the solutions are integers or fractions without square roots).

If only 2 terms → Use basic factorization by finding the HCF.

If only 2 terms and is also difference of squares → Use the a^2 - b^2 = (a - b) (a + b) formula to factorize.

If monic quadratic expression with 3 terms → Use inspection (faster method) to find 2 numbers that multiply to c and add to b. Only factorize by grouping if inspection doesn’t work for you.

If non-monic quadratic expression with 3 terms →

* If coefficient of x^2 (a) is a common factor of a, b, and c → factorize it out of the quadratic expression, and factorize the remaining monic quadratic expression.
* If coefficient of x^2 (a) is not a common factor of a, b, and c → find 2 numbers that multiply to ac and add to b, then factorize by grouping.

Algebraic fractions

Fractions with an algebraic expression in the numerator and/or denominator.

How do you simplify algebraic fractions?

1. Fully factorize the numerator and denominator.
2. Cancel out any common factors in the numerator and denominator.

Example: Simplify (3x + 6) / (4x + 8)

(3x + 6) / (4x + 8) = \[3(x + 2)\] / \[4(x + 2)\] = 3/4

How do you add and subtract algebraic fractions?

1. Find the lowest common denominator.
2. Multiply the numerator and denominator of the fractions by their relevant factor so the denominator becomes the lowest common denominator without changing the fraction’s value (i.e. find an equivalent fraction where the denominator is the lowest common denominator).
3. Add or subtract the numerators and divide this result by the lowest common denominator to get a singular fraction.
4. Keep the numerators and denominators in factorized form so common factors can be cancelled out.

Example: Simplify x/(x + 4) - 3/(x - 1)

x/(x + 4) - 3/(x - 1) = \[x(x-1)\]/\[(x+4)(x-1)\] - \[3(x+4)\]/\[(x+4)(x-1)\] = (x^2 - x)/\[(x+4)(x-1)\] - (3x + 12)/\[(x+4)(x-1)\] = (x^2 - 4x - 12)/\[(x+4)(x-1)\] = \[(x-6)(x+2)\]/\[(x-1)(x+4)\]

How do you multiply algebraic fractions?

1. Fully factorize the numerators and denominators.
2. Cancel out any common factors in the numerators and denominators.
3. Multiply the numerators together.
4. Multiply the denominators together.
5. The final result is the product from step 3 as the numerator divided by the product from step 4 as the denominator.
6. Cancel out any common factors (if any) to completely simplify the fraction obtained from step 5.

How do you divide algebraic fractions?

1. Dividing by a fraction a/b is the same as multiplying by that fraction’s reciprocal. (Only do this to the algebraic fraction written AFTER the division sign):


1. ➗ a/b is the same as x b/a.
2. Multiply the algebraic fraction before the division sign to the reciprocal of the algebraic fraction after the division sign.

How do you solve algebraic fractions?

1. Multiply the entire equation by the lowest common denominator. Ensure you multiply with brackets.
2. Expand the brackets on both sides of the equation.
3. Simplify both sides of the equation by collecting like terms.
4. Rearrange the equation so all variables are on the left hand side and all constants are on the right hand side (or so that it takes the form of a solvable linear or quadratic equation).
5. Solve the equation.
6. Remember that when you multiply everything by the lowest common denominator, the lowest common denominator cannot be 0, so the variable cannot have any value that causes the denominator to be 0 otherwise the entire algebraic fractions’ values would be undefined.

Example: Solve 4/(x-3) + 5/(x+1) = 5

First multiply everything by the lowest common denominator which is (x-3) (x+1), now x ≠ 3 and x ≠ -1 or else the denominator would be 0:

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

Expand the brackets and simplify.

4x + 4 + 5x - 15 = 5(x^2 - 2x - 3)

9x - 11 = 5x^2 - 10x - 15

Rearrange into a quadratic equation format.

5x^2 - 19x - 4 = 0

Solve the quadratic equation:

(5x + 1)(x - 4) = 0

5x + 1 = 0 OR x - 4 = 0

x = -1/5 OR x = 4

Completing the square

Process of writing a quadratic in the form of a perfect square added to a constant where necessary for ease of solving. The quadratic ax^2 + bx + c = 0 will be written as:

a(x+p)^2 + q = 0

How do you complete the square for quadratic equations/expressions?

1. Write the first 2 terms of the quadratic expression as a difference of squares:
2. ax^2 + bx + c can be written as:


1. a \[x^2 + (b/a)x\] + c =
2. a \[(x + b/2a)^2 - (b/2a)^2\] + c =
3. a(x + (b/2a))^2 - a(b/2a)^2 + c =
4. a(x + (b/2a))^2 - b^2/4a + c
5. Simplify - b^2/4a + c to get a single constant then add or subtract that to a(x + (b/2a))^2 to get the final answer.
6. If it’s a quadratic equation, you can solve for the variable’s value.
7. If it’s a quadratic expression, simply leave it as it is in this simplest factorized form.

Example: Solve 4x^2 + 16x + 5 by completing the square.

4x^2 + 16x + 5 =

4\[x^2 + 4x\] + 5 =

4\[(x+2)^2 - 2^2\] + 5 =

4(x+2)^2 - 4(2^2) + 5 =

4(x+2)^2 - 16 + 5 =

4(x+2)^2 - 11 = 0

We have now completed the square and can solve the equation:

4(x+2)^2 - 11 = 0

4(x+2)^2 = 11

(x+2)^2 = 11/4

x + 2 = ± √(11/4) = ± (1/2)√11

x = -2 ± (1/2)√11

Vertex

Turning point of the quadratic graph. It’s the maximum point if a < 0 in ax^2 + bx + c (graph is an n-shaped parabola that extends downwards), it’s the minimum point if a > 0 in ax^2 + bx + c (graph is a u-shaped parabola that extends upwards).

What is the turning point of the graph y = x^2

0,0 (the origin).

How do you find the turning point after completing the square for a quadratic expression?

After completing the square for a quadratic expression and getting: y = a(x + (b/2a))^2 - b^2/4a + c

y-value of vertex = - b^2/4a + c (because it’s the same as translating -b^2/4a + c units up (if positive) or down (if negative) from the vertex of y = x^2).

x-value of vertex = -b/2a (because it’s the same as translating b/2a units leftwards from the vertex of y = x^2).

x-value of vertex can be found by finding the average of the 2 x-intercepts where the parabola of the quadratic graph intersects the x-axis (where y = 0):

* 0 = a(x + (b/2a))^2 - b^2/4a + c
* a(x + (b/2a))^2 = b^2/4a - c = (b^2 - 4ac)/4a
* (x + (b/2a))^2 = (b^2 - 4ac)/(4a^2)
* x + (b/2a) = ± √\[(b^2 - 4ac)/(4a^2)\] = ± \[√(b^2 - 4ac)\]/2a
* x = -b/2a ± \[√(b^2 - 4ac)\]/2a = \[-b ± √(b^2 - 4ac)\]/2a
* x1 = \[-b - √(b^2 - 4ac)\]/2a
* x2 = \[-b + √(b^2 - 4ac)\]/2a
* x1 + x2 = \[-b - √(b^2 - 4ac)\]/2a + \[-b + √(b^2 - 4ac)\]/2a = -2b/2a = -b/a
* x1 + x2 = -b/2a = x-value of vertex

Therefore, for any quadratic graph of the equation y = a(x + (b/2a))^2 - b^2/4a + c, the vertex will be **(-b/2a, - b^2/4a + c)**

What can you do when given a quadratic equation’s turning point?

You can create/find the quadratic equation when given its turning point.

You can use the fact that all squared terms/expressions are nonnegative to set limits to results using inequalities. For example:

y = x^2 + 6x - 3

y = (x + 3)^2 - 12

(x + 3)^2 ≥ 0 →

y ≥ -12

Subject (meaning in formulae)

The variable isolated on its own to 1 side of the equation or formula, it is the variable being calculated.

How to rearrange formulae and change the subject?

Conduct inverse operations to isolate the variable you want to change into the subject. If powers are involved, you can use roots to inverse their effect.

Example: Rearrange the formula v^2 = u^2 + 2as so that u is the subject.

v^2 = u^2 + 2as

u^2 = v^2 - 2as

u = √(v^2 - 2as)

If the subject appears twice, rearrange all the terms containing the variable to 1 side of the equation, and all of the terms not containing that variable to the other side of the equation. Factorize out the variable, then divide the right hand side by the bracketed expression.

For example: Rearrange p = (2 - ax) / (x - b) so x is the subject.

p(x - b) = 2 - ax

px - pb = 2 - ax

px + ax = pb + 2

x (p + a) = pb + 2

x = (pb + 2) / (p + a)

Algebraic proof

Proof which uses algebra to show that something is true in every case.

Proving about odd and even numbers.

Even numbers must be in the form of 2n;

Odd numbers must be in the form of 2n + 1;

where n is an integer in both cases.

It’s a good idea to write a sentence at the end of your algebraic proof to state what has been proved (usually just copy the statement that needs to be proved from the question).