Factorise x² - x: A Step-by-Step Guide

Factorising the quadratic expression \( x^2 - x \) involves finding the common factors of the terms in the expression. Here’s a step-by-step guide:

Step 1: Identify the common factor

In the expression \( x^2 - x \), the common factor is \( x \).

Step 2: Factor out the common factor

Once the common factor is identified, factor it out of each term:

\[ x^2 - x = x(x - 1) \]

Step 3: Verify the factorisation

To ensure the factorisation is correct, expand the factored expression to check if it matches the original expression:

\[ x(x - 1) = x^2 - x \]

Since the expanded form is the same as the original expression, the factorisation is correct.

Conclusion

The factorised form of \( x^2 - x \) is \( x(x - 1) \). This process can be used to factorise similar quadratic expressions by identifying and factoring out the common factor.

Examples

• Example 1: Factorise \( x^2 - 2x \)

\[ x^2 - 2x = x(x - 2) \]

• Example 2: Factorise \( x^2 - 3x \)

\[ x^2 - 3x = x(x - 3) \]

Further Reading

For more information on factorising quadratic expressions, you can refer to various algebra textbooks or online resources such as Khan Academy and Mathway.

Factorisation is a fundamental concept in algebra that involves breaking down a complex expression into simpler components, known as factors, that when multiplied together give the original expression. This process is essential for solving polynomial equations and simplifying expressions.

To factorise the quadratic expression \(x^2 - x\), we follow these steps:

1. Identify the common factor: In the expression \(x^2 - x\), both terms share a common factor of \(x\).
2. Factor out the common factor: We factor \(x\) out of each term to get \(x(x - 1)\).
3. Verify the factors: By multiplying the factors back together, \(x(x - 1)\), we confirm that the original expression \(x^2 - x\) is obtained.

Thus, the factorised form of \(x^2 - x\) is \(x(x - 1)\).

Understanding how to factorise expressions like \(x^2 - x\) is crucial for solving equations, simplifying expressions, and finding roots of polynomials. Mastery of factorisation provides a strong foundation for more advanced mathematical concepts.

Quadratic expressions are algebraic expressions where the highest power of the variable is two. The general form of a quadratic expression is \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. Understanding how to factor these expressions is crucial in solving quadratic equations.

Let's consider the expression \( x^2 - x \). This can be factored as follows:

1. Identify the common factor in each term. In this case, both terms \( x^2 \) and \( -x \) have a common factor of \( x \).
2. Factor out the common factor \( x \): \( x^2 - x = x(x - 1) \).

This shows that \( x^2 - x \) can be written as \( x(x - 1) \). This is the factored form of the quadratic expression.

Quadratic expressions can be factored using different techniques, such as:

• Factoring by grouping: This involves grouping terms with common factors and then factoring out the greatest common factor (GCF) from each group.
• Using the quadratic formula: For expressions that cannot be easily factored, the quadratic formula \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \) can be used to find the roots of the quadratic equation, which then helps in factoring the expression.
• Completing the square: This method involves rewriting the quadratic expression in the form \( (x + p)^2 = q \) and then solving for \( x \).

By mastering these techniques, you can handle a wide range of quadratic expressions and solve related equations effectively.

Factorising quadratic expressions is a fundamental algebraic skill. To factorise x² - x, follow these basic steps:

1. Identify the common factor in each term of the quadratic expression. In x² - x, the common factor is x.
2. Factor out the common factor from each term. This means we write the expression as a product of the common factor and another expression.
3. Express x² - x as x(x - 1). This is because x² divided by x is x and -x divided by x is -1.

Therefore, the factorised form of x² - x is x(x - 1).

Here is the step-by-step process in MathJax format:

\[ x^2 - x = x(x - 1) \]

Factorising simplifies the equation and is crucial for solving quadratic equations, finding roots, and simplifying expressions.

Identifying common factors is a crucial step in factorising expressions. In algebra, a common factor is a term that is present in each part of an expression. Here's a step-by-step guide to identifying and using common factors in the expression \(x^2 - x\):

1. Examine Each Term: Look at each term in the expression. In \(x^2 - x\), we have two terms: \(x^2\) and \(-x\).
2. Identify the Common Factor: Determine the greatest common factor (GCF) that is shared by each term. Both \(x^2\) and \(-x\) have a common factor of \(x\).
3. Factor Out the Common Factor: Write the expression as a product of the common factor and another term. For \(x^2 - x\), factor out \(x\):

   \[
   x^2 - x = x(x - 1)
   \]

4. Verify the Factorisation: Distribute the common factor to ensure the original expression is obtained:

   \[
   x(x - 1) = x^2 - x
   \]

By following these steps, we see that the expression \(x^2 - x\) can be successfully factorised as \(x(x - 1)\). Identifying and factoring out common factors simplifies expressions and is a key skill in solving algebraic equations.

Factoring out the common factor is a fundamental step in simplifying expressions and solving equations. Let's take a detailed look at how to factor out the common factor in the quadratic expression \(x^2 - x\).

Given the expression \(x^2 - x\), we follow these steps:

1. Identify the common factor in each term of the expression. Here, both terms, \(x^2\) and \(-x\), contain the variable \(x\).
2. Factor out the common factor \(x\) from each term. This involves dividing each term by \(x\) and placing the common factor outside a set of parentheses:

\[
x^2 - x = x(x - 1)
\]

Thus, the expression \(x^2 - x\) is factored into \(x(x - 1)\). This process simplifies the expression and is crucial for solving quadratic equations and other algebraic manipulations.

To further understand this method, consider the following table summarizing the steps:

By understanding and applying this method, we can simplify many quadratic expressions efficiently.

Once you have factorised an expression, it is crucial to verify the factorisation to ensure accuracy. Verification involves expanding the factors to see if you retrieve the original expression. Here are the steps to verify the factorisation of \(x^2 - x\).

1. Start with the factorised form of the expression. For \(x^2 - x\), the factorised form is \(x(x - 1)\).

2. Expand the factors:

   • \(x(x - 1) = x^2 - x\)

3. Compare the expanded form with the original expression:

   • The expanded form \(x^2 - x\) matches the original expression \(x^2 - x\).

If the expanded form matches the original expression, the factorisation is verified. If not, recheck the factorisation steps.

Factorisation is a method used to express a mathematical expression as a product of its factors. Let's explore several examples that follow a similar pattern to factorising x² - x.

1. Factorising x² - 4

• This expression is a difference of squares. We can factorise it using the identity a² - b² = (a - b)(a + b).
• Here, a is x and b is 2. Thus, x² - 4 can be factorised as:
  • (x - 2)(x + 2)

2. Factorising x² - 9x

• This expression can be factorised by identifying the common factor x.
• We can factor out x from each term:
  • x(x - 9)

3. Factorising 3x² - 12x

• This expression has a common factor of 3x.
• We can factor out 3x from each term:
  • 3x(x - 4)

4. Factorising x² - 5x + 6

• This quadratic expression can be factorised by finding two numbers that multiply to 6 (the constant term) and add to -5 (the coefficient of the linear term).
• The numbers -2 and -3 satisfy these conditions. So, we can write:
  • x² - 5x + 6 = (x - 2)(x - 3)

5. Factorising x² + 7x + 12

• We need to find two numbers that multiply to 12 (the constant term) and add to 7 (the coefficient of the linear term).
• The numbers 3 and 4 satisfy these conditions. So, we can write:
  • x² + 7x + 12 = (x + 3)(x + 4)

6. Factorising x² + 4x - 12

• To factorise this quadratic expression, find two numbers that multiply to -12 and add to 4.
• The numbers 6 and -2 work. Thus, we can write:
  • x² + 4x - 12 = (x + 6)(x - 2)

These examples illustrate various techniques for factorising quadratic expressions, including identifying common factors and using algebraic identities.

In this section, we will explore advanced techniques for factorising quadratic expressions. These techniques go beyond the basic methods and can be particularly useful for more complex expressions or when a straightforward factorisation is not immediately apparent.

1. Completing the Square

Completing the square is a method that involves transforming a quadratic expression into a perfect square trinomial. This technique is useful for solving quadratic equations and for understanding the properties of quadratic functions.

1. Start with the quadratic expression: \( ax^2 + bx + c \).
2. Divide all terms by \( a \) (if \( a \neq 1 \)) to simplify: \( x^2 + \frac{b}{a}x + \frac{c}{a} \).
3. Rewrite the expression by adding and subtracting the square of half the coefficient of \( x \):

   \( x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2 + \frac{c}{a} \).

4. Combine the first three terms into a perfect square trinomial and simplify:

   \( \left(x + \frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2 + \frac{c}{a} \).

5. Factorise the resulting expression:

   \( \left(x + \frac{b}{2a}\right)^2 - \frac{b^2 - 4ac}{4a^2} \).

2. Using the Quadratic Formula

The quadratic formula can be used to find the roots of a quadratic equation, which can then help in factorising the expression.

1. For the quadratic equation \( ax^2 + bx + c = 0 \), the roots are given by:

   \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

2. Once the roots (\( r_1 \) and \( r_2 \)) are found, the quadratic expression can be factorised as:

   \( a(x - r_1)(x - r_2) \).

3. Using Algebraic Identities

Algebraic identities can simplify the factorisation process for specific types of quadratic expressions. Here are some useful identities:

• Difference of squares: \( a^2 - b^2 = (a - b)(a + b) \).
• Perfect square trinomial: \( a^2 \pm 2ab + b^2 = (a \pm b)^2 \).
• Sum and difference of cubes (useful for higher-degree polynomials but sometimes applicable in quadratic forms).

4. Using Synthetic Division

Synthetic division is a simplified form of polynomial division, particularly useful for dividing by linear factors.

1. Identify a possible root of the quadratic expression \( ax^2 + bx + c \).
2. Use synthetic division to divide the quadratic expression by \( x - \text{root} \).
3. If the division results in a remainder of 0, the divisor is a factor.

Example: Factorising \( x^2 - 4 \)

Let's apply these techniques to factorise the expression \( x^2 - 4 \).

1. Recognise \( x^2 - 4 \) as a difference of squares:

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

By using these advanced factorisation techniques, you can handle a wide range of quadratic expressions efficiently. Practice these methods to become proficient in factorising complex quadratics.

When factorising expressions like \( x^2 - x \), algebraic identities play a crucial role in simplifying and finding the factors efficiently.

One powerful identity to utilize here is:

• The difference of squares identity: \( a^2 - b^2 = (a - b)(a + b) \).

Applying this identity to \( x^2 - x \), we can rewrite it as:

This factorisation is derived by recognizing \( x^2 - x \) as a difference of squares where \( a = x \) and \( b = 1 \).

Another useful identity is:

• The distributive property: \( a(b + c) = ab + ac \).

Using the distributive property, we can verify the factorisation:

Thus, confirming that \( x(x - 1) \) correctly factors \( x^2 - x \).

Understanding the graphical representation of factorised forms like \( x(x - 1) \) for \( x^2 - x \) helps visualize how the expression behaves.

Consider the following steps to interpret the graph:

1. Plot the function \( y = x^2 - x \) initially.
2. Identify key points such as intercepts and vertex.
3. Factorise \( x^2 - x \) to \( x(x - 1) \).
4. Plot the simplified form \( y = x(x - 1) \).
5. Observe how the factorised form relates to the original graph.

Benefits of understanding the graphical representation include:

• Clarity in identifying roots and behavior.
• Visualization aids in comprehending factorisation.
• Facilitates solving equations and inequalities graphically.

The factorisation of expressions such as \( x^2 - x \) has practical applications in solving equations in algebra.

Follow these steps to apply factorisation in solving equations:

1. Identify the equation involving \( x^2 - x \).
2. Factorise the expression \( x^2 - x \) into \( x(x - 1) \).
3. Set each factor equal to zero and solve for \( x \):
   • Solve \( x = 0 \).
   • Solve \( x - 1 = 0 \), which gives \( x = 1 \).
4. Verify solutions by substituting back into the original equation.

Applications of solving equations using factorisation include:

• Efficiently finding roots of quadratic equations.
• Streamlining the process of solving algebraic equations.
• Understanding how factors affect the solutions graphically.

Factorising expressions like \( x^2 - x \) can sometimes lead to common mistakes if not approached carefully. Here are some pitfalls to watch out for:

• Forgetting to factor out the greatest common factor: Always check if there is a common factor that can be extracted first.
• Incorrect application of algebraic identities: Ensure that the identities used are appropriate for the expression being factorised.
• Misidentifying the correct form: Mistaking the expression for a different form can lead to incorrect factorisation.
• Errors in distributing or combining terms: Pay attention to each step of the factorisation process to avoid calculation errors.

To avoid these mistakes, follow these steps:

1. Review the expression carefully before attempting to factorise.
2. Verify your factorisation by expanding back to the original expression.
3. Practice using different methods and identities to reinforce understanding.

Practice problems help reinforce the understanding of factorising expressions such as \( x^2 - x \). Here are some practice problems along with their solutions:

1. Factorise \( x^2 - x \).
2. Factorise \( x^2 - 2x \).
3. Factorise \( x^2 + 3x \).
4. Factorise \( 2x^2 - 5x \).
5. Factorise \( 3x^2 + 6x \).

Solutions:

1. \( x^2 - x = x(x - 1) \).
2. \( x^2 - 2x = x(x - 2) \).
3. \( x^2 + 3x = x(x + 3) \).
4. \( 2x^2 - 5x = x(2x - 5) \).
5. \( 3x^2 + 6x = 3x(x + 2) \).

Expand your understanding of factorising quadratic expressions with these additional resources:

Factorising quadratic expressions like \( x^2 - x \) is a fundamental skill in algebra that enables us to simplify and solve equations efficiently.

Key takeaways from this guide include:

• Understanding the basic steps of factorisation.
• Identifying and applying common algebraic identities.
• Utilising factorisation in solving equations and inequalities.
• Visualising factorised forms graphically to comprehend their behavior.
• Avoiding common mistakes through careful review and practice.

By mastering these concepts, you can enhance your problem-solving abilities and approach more complex algebraic challenges with confidence.