Quadratic Expressions

An algebraic expression having the degree of '2' is called a quadratic expression. By 'degree', we mean the highest exponent on the variable/s in an algebraic expression.

For example, `x^2 + 2x + 3` is an algebraic expression in which the variable `x` has the highest exponent of `2`.

Quadratic expressions can be written in various forms, namely the General Form, the Vertex Form and the Intercept Form. A quadratic expression can be converted from one form to another by using algebraic operations.

A quadratic expression can be factored into a product of two linear expressions by the method of splitting the middle term.
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Introduction to Quadratics

Quadratics is the branch of mathematics which deals with factoring quadratic expressions, solving quadratics equations and graphing quadratic functions.

A quadratic expression is an algebraic expression having the degree of 2. It is typically an expression in the variable 'x' (or any other variable) having the highest exponent of 2. An example is
`x^2 + 2x+ 3`
A quadratic equation is an equation in which we equate a quadratic expression with zero. Thus, the following is a quadratic equation:
`x^2 + 2x + 3 = 0`
A quadratic equation does not have more than two solutions. They are called the roots, zeros or x-intercepts of the quadratic equation.

A quadratic function is the equation of two variables in which the dependent variable is equal to a quadratic expression in the independent variable. For example, in the following equation:
`y = x^2 + 2x + 3`
the dependent variable is 'y' and it is equal to a quadratic expression in the independent variable 'x'.
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Convert quadratic function from standard to vertex form

The standard form of a quadratic equation is as follows:
`ax^2 + bx + c = 0`
It's vertex form is as follows:
`a(x- h)^2 + k = 0`
A quadratic equation can be written in both forms. It can be converted from the standard to the vertex form by the method of completing the square.

The following example explains the method.

Consider the quadratic equation in general (standard) form:
`x^2 + 2x + 3 = 0`
We will convert it to vertex form by the following steps:

Move the constant term to the right hand side

The constant term (number without a variable) in the above equation is '3'. We can move it to the right hand side of the = sign as follows:
`x^2 + 2x = -3`

Identify the coefficient of 'x'

The coefficient (number near) 'x' in the above equation is '2'.

Divide the number by 2

Dividing the number '2' by 2, we get 1.

Complete the square

Notice that `2x` in the equation has a `+` sign before it. Thus, we will take the number obtained in the previous step and add it with 'x' and then square the expression as follows:
`(x + 1)^2`
Now we can replace the left hand side of the quadratic equation with the above expression as follows:
`(x + 1)^2 = -3`

Move the constant term in the right hand side again to the left hand side of the equation

`(x + 1)^2 + 3 = 0`
The above equation is said to be the vertex form of `x^2 + 2x + 3 = 0`
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Favorable Outcomes

Favorable outcomes in probability are those results of an experiment which you want.

Favorable outcomes of a coin

Suppose two friends toss a coin. Friend A wants 'heads' and B wants 'tails'. We can say that the favorable outcome for A is 'heads' and for B is 'tails'.

When a coin is tossed, it can result in either heads or tails. Thus, there are two possible outcomes on tossing a coin. Any one of these outcomes, heads or tails, can be favorable according to what you want.

If suppose someone says they want either heads or tails, then the favorable outcomes will be 2, because whether the coin gives heads or tails the outcome is what they wanted.

Favorable outcomes of a number cube

A number cube has six faces numbered 1 through 6 and on throwing it only one of these faces shows up.

Suppose you have to calculate the probability of getting an even number on rolling a number cube. The number cube has the numbers 1, 2, 3, 4, 5 and 6 on it, out of which only 2, 4 and 6 are even numbers. Thus, there are a total of 6 possible outcomes but only 3 favorable outcomes. If any one of these favorable outcomes occurs, then you say that the result is favorable, or of interest.
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Solved Examples: Difference of Cubes

Solved Example 1

Factor `x^3 - 8`

Solution

Rewrite 8 as `2^3`,
`x^3 - 2^3`
Comparing with `a^3 - b^3`, we get
`a = x` and `b = 2`
Applying the formula of difference of cubes: `a^3 - b^3 = (a - b)(a^2 + ab + b^2)`, we get
`(x - 2)(x^2 + x*2 + 2^2)`
Simplifying,
`(x - 2)(x^2 + 2x + 4)`

Solved Example 2

Factor `8a^3 - b^3`

Solution

Rewrite 8 as `2^3`,
`(2a)^3 - b^3`
Comparing with `a^3 - b^3`, we get,
`a = 2a` and `b = b`
Applying the formula `a^3 - b^3 = (a - b)(a^2 + ab + b^2)`, we get
`(2a - b)((2a)^2 + (2a)(b) + b^2)`
Simplifying,
`(2a - b)(4a^2 + 2ab + b^2)`

Solved Example 3

Factor `1 - x^3`

Solution

Rewrite 1 as `1^3` (because `1^3 = 1 * 1 * 1 = 1`).
`1^3 - x^3`
Comparing with `a^3 - b^3`, we get
`a = 1` and `b = x`
Applying the formula `a^3 - b^3 = (a - b)(a^2 + ab + b^2)`, we get
`(1 - x)(1^2 + 1*x + x^2)`
Simplifying,
`(1 - x)(1 + x + x^2)`
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Worksheet: Difference of Squares

Instructions

Click on Show Answer button to show a particular answer. Click on the answer itself to hide it. Use the buttons at the bottom to show/hide all answers
Question
Answer
`x^2 - 16`
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`x^4 - 16`
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`2x^2 - 50`
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`x^3 - xy^2`
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`81x^4 - y^4`
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`125x^3 - 36x`
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`162 - 8y^2`
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Solved Examples: Difference of Squares

Solved Example 1

Factor `x^2 - 4`

Solution

Rewrite 4 as `2^2`
`x^2 – 4 = x^2 – 2^2`
Comparing with `a^2 – b^2`, we get `a = x` and `b = 2`. Applying the formula `a^2 – b^2 = (a + b)(a – b)`, we get
`x^2 – 2^2 = (x + 2)(x – 2)`

Solved Example 2

Factor `x^4 – 81`

Solution

Rewrite `x^4` as `(x^2)^2` and 81 as `9^2`. Thus, we get
`x^4 – 81 = (x^2)^2 – 9^2`
Comparing with `a^2 – b^2`, we get `a = x^2` and `b = 9`. Applying the formula `a^2 – b^2 = (a + b)(a – b)`, we get
`(x^2)^2 – 9^2 = (x^2 + 9)(x^2 – 9)`
Rewrite 9 as `3^2`
`(x^2 + 9)(x^2 – 9) = (x^2 + 9)(x^2 – 3^2)`
Again applying the formula `a^2 – b^2 = (a + b)(a – b)`, we get
`(x^2 + 9)(x^2 – 3^2) = (x^2 + 9)(x + 3)(x – 3)`

Solved Example 3

Factor `2x^2 – 72`

Solution

Factor out 2 from the given expression,
`= 2(x^2 – 36)`
Rewrite 36 as `6^2`.
` = 2(x^2 – 6^2)`
Applying the formula `a^2 – b^2 = (a + b)(a – b)`, we get
`= 2(x + 6)(x – 6)`

Solved Example 4

Factor `a^2 – b^4`

Solution

Rewrite `b^4` as `(b^2)^2`  to get
`a^2 – (b^2)^2`
Applying the formula `a^2 – b^2 = (a + b)(a – b)`, we get
`(a + b^2)(a – b^2)`

Solved Example 5

Factor `25x^2 – 36y^2`

Solution

Rewrite 25 as `5^2` and 36 as `6^2`.
`(5x)^2 – (6y)^2`
Applying the formula `a^2 – b^2 = (a + b)(a – b)`, we get
`(5x + 6y)(5x – 6y)`

Solved Example 6

Factor `32x^3 – 50x`

Solution

Factor out `2x` from the given expression,
`= 2x(16x^2 – 25)`
Rewrite 16 as `4^2` and 25 as `5^2`
`= 2x((4x)^2 – 5^2)`
Applying the formula `a^2 – b^2 = (a + b)(a – b)`,
`= 2x(4x + 5)(4x – 5)`
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Worksheet: Cube of a Difference

Instructions

Click on Show Answer button to show a particular answer. Click on the answer itself to hide it. Use the buttons at the bottom to show/hide all answers
Question
Answer
`(4x - y)^3`
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`(5x - 4b)^3`
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`(a/3 + b)^3`
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`(x/2 + 1/y)^3`
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`(3x - 2y)^3`
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`(1 - x)^3`
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`(y - 2)^3`
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`(a/3 + 2/b)^3`
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Solved Examples: Cube of a Difference `(a - b)^3`

Solved Example 1

Expand `(x -2y)^3`

Solution

Step 1: Compare the given expression with `(a - b)^3`

Comparing  `(x - 2y)^3` with `(a - b)^3`, we get
`a = x` and `b = 2y`

Step 2: Apply the formula `(a - b)^3 = a^3 - b^3 - 3ab(a - b)`

Putting `x` in place of `a` and `2y` in place of `b` in the above formula, we get,
`(x - 2y)^3 = x^3 - (2y)^3 - 3(x)(2y)(x - 2y)`
Simplifying the left hand expression,
`(x - 2y)^3 = x^3 - 8y^3 - 6xy(x - 2y)`
`= x^3 - 8y^3 - 6x^2y + 12xy^2`

Solved Example 2

Expand `(3x -2y)^3`

Solution

Step 1: Compare the given expression with `(a - b)^3`

Comparing  `(3x - 2y)^3` with `(a - b)^3`, we get
`a = 3x` and `b = 2y`

Step 2: Apply the formula `(a - b)^3 = a^3 - b^3 - 3ab(a - b)`

Putting `3x` in place of `a` and `2y` in place of `b` in the above formula, we get,
`(3x - 2y)^3 = (3x)^3 - (2y)^3 - 3(3x)(2y)(3x - 2y)`
Simplifying the left hand expression,
`(3x - 2y)^3 = 27x^3 - 8y^3 - 18xy(3x - 2y)`
`= 27x^3 - 8y^3 - 54x^2y + 36xy^2`

Solved Example 3

Expand `(x/3 - 2y)^3`

Solution

Step 1: Compare the given expression with `(a - b)^3`

Comparing  `(x/3 - 2y)^3` with `(a - b)^3`, we get
`a = x/3` and `b = 2y`

Step 2: Apply the formula `(a - b)^3 = a^3 - b^3 - 3ab(a - b)`

Putting `x/3` in place of `a` and `2y` in place of `b` in the above formula, we get,
`(x/3 - 2y)^3 = (x/3)^3 - (2y)^3 - 3(x/3)(2y)(x/3 - 2y)`
Simplifying the left hand expression,
`(x/3 - 2y)^3 = (x^3)/27 - 8y^3 - 2xy(x - 2y)`
`= (x^3)/27 - 8y^3 - 2x^2y + 4xy^2`

Solved Example 4

Expand `(a/2 - b/3)^3`

Solution

Step 1: Compare the given expression with `(a - b)^3`

Comparing  `(a/2 - b/3)^3` with `(a - b)^3`, we get
`a = a/2` and `b = b/3`

Step 2: Apply the formula `(a - b)^3 = a^3 - b^3 - 3ab(a - b)`

Putting `a/2` in place of `a` and `b/3` in place of `b` in the above formula, we get,
`(a/2 - b/3)^3 = (a/2)^3 - (b/3)^3 - 3(a/2)(b/3)(a/2 - b/3)`
Simplifying the left hand expression,
`(a/2 - b/3)^3 = (a^3)/8 - (b^3)/27 - (ab)/2(a/2 - b/3)`
`= (a^3)/8 - (b^3)/27 - (a^2b)/4 + (ab^2)/6`

Solved Example 5

Expand `(x - 1)^3`

Solution

Step 1: Compare the given expression with `(a - b)^3`

Comparing  `(x - 1)^3` with `(a - b)^3`, we get
`a = x` and `b = 1`

Step 2: Apply the formula `(a - b)^3 = a^3 - b^3 - 3ab(a - b)`

Putting `x` in place of `a` and `1` in place of `b` in the above formula, we get,
`(x - 1)^3 = x^3 - 1^3 - 3(x)(1)(x - 1)`
Simplifying the left hand expression,
`(x - 1)^3 = x^3 - 1 - 3x(x - 1)`
`= x^3 - 1 - 3x^2 + 3x`
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Worksheet: Sum of a Cube `(a + b)^3`

Instructions

Click on Show Answer button to show a particular answer. Click on the answer itself to hide it. Use the buttons at the bottom to show/hide all answers
Question
Answer
`(2x + y)^3`
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`(x + 3y)^3`
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`(3x + 2y)^3`
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`(4a + 7b)^3`
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`(x/2 + 1)^3`
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`(x/3 + y)^3`
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`(x/2 + y/3)^3`
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`(a/3 + b/4)^3`
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`((2x)/3 + (3y)/4)^3`
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`((3a)/2 + (4a)/3)^3`
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