Factorise X 2 10x 16

Factorising Quadratic Expressions: A Deep Dive into x² + 10x + 16

This article provides a comprehensive guide to factorising the quadratic expression x² + 10x + 16. We'll explore different methods, delve into the underlying mathematical principles, and build your understanding of quadratic equations. This will equip you with the skills to tackle similar problems with confidence. Understanding quadratic factorisation is crucial for many areas of mathematics, including algebra, calculus, and beyond. Let's get started!

Understanding Quadratic Expressions

Before we delve into factorising x² + 10x + 16, let's understand what a quadratic expression is. A quadratic expression is an algebraic expression of the second degree, meaning the highest power of the variable (usually x) is 2. It generally takes the form ax² + bx + c, where 'a', 'b', and 'c' are constants (numbers). In our case, a = 1, b = 10, and c = 16.

Method 1: Factorising by Inspection (Trial and Error)

This method involves finding two numbers that add up to 'b' (the coefficient of x) and multiply to 'c' (the constant term). Let's apply this to x² + 10x + 16:

1. Find the factors of 'c': The factors of 16 are 1 and 16, 2 and 8, and 4 and 4.

2. Identify the pair that adds up to 'b': We need two numbers that add up to 10. The pair 2 and 8 satisfies this condition (2 + 8 = 10).

3. Write the factorised form: Since the coefficient of x² is 1, we can directly write the factorised form as (x + 2)(x + 8).

Therefore, x² + 10x + 16 = (x + 2)(x + 8). We can verify this by expanding the brackets: (x + 2)(x + 8) = x² + 8x + 2x + 16 = x² + 10x + 16.

Method 2: Completing the Square

This method involves manipulating the quadratic expression to form a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the square of a binomial. The general form is (ax + b)².

Let's apply this to x² + 10x + 16:

1. Focus on the x² and x terms: Consider only x² + 10x.

2. Find half of the coefficient of x: Half of 10 is 5.

3. Square the result: 5² = 25.

4. Add and subtract the result: We add and subtract 25 to maintain the value of the expression: x² + 10x + 25 - 25 + 16.

5. Form the perfect square trinomial: The first three terms (x² + 10x + 25) form a perfect square trinomial, which is (x + 5)².

6. Simplify the remaining terms: -25 + 16 = -9.

7. Rewrite the expression: The expression becomes (x + 5)² - 9.

8. Factor the difference of squares: This is in the form a² - b², which factors to (a + b)(a - b). Here, a = (x + 5) and b = 3. Therefore, (x + 5)² - 9 = (x + 5 + 3)(x + 5 - 3) = (x + 8)(x + 2).

Thus, x² + 10x + 16 = (x + 2)(x + 8), the same result as before. Completing the square is a more general method that works even when the coefficient of x² is not 1.

Method 3: Using the Quadratic Formula

The quadratic formula is a powerful tool for finding the roots (solutions) of any quadratic equation of the form ax² + bx + c = 0. The roots are given by:

x = [-b ± √(b² - 4ac)] / 2a

While this directly gives the roots, it can be used to find the factors. For x² + 10x + 16 = 0, a = 1, b = 10, and c = 16. Substituting these values into the quadratic formula:

x = [-10 ± √(10² - 4 * 1 * 16)] / 2 * 1

x = [-10 ± √(100 - 64)] / 2

x = [-10 ± √36] / 2

x = [-10 ± 6] / 2

This gives two solutions:

x₁ = (-10 + 6) / 2 = -2

x₂ = (-10 - 6) / 2 = -8

The factors are then (x - x₁) and (x - x₂), which are (x - (-2)) = (x + 2) and (x - (-8)) = (x + 8). Therefore, the factorised form is (x + 2)(x + 8).

Visualizing the Quadratic Expression: Graphing

Graphing the quadratic equation y = x² + 10x + 16 provides a visual representation of the expression. The x-intercepts (where the graph crosses the x-axis) represent the roots of the equation (where y = 0). These x-intercepts directly correspond to the values we found using the quadratic formula (-2 and -8). The factors are then easily derived from these roots.

Why Factorisation is Important

Factorising quadratic expressions is a fundamental skill in algebra with numerous applications:

• Solving Quadratic Equations: Factorisation allows us to find the roots (solutions) of quadratic equations easily. Setting each factor to zero and solving gives the roots.

• Simplifying Algebraic Expressions: Factorisation helps simplify complex expressions, making them easier to work with.

• Graphing Quadratic Functions: The factorised form quickly reveals the x-intercepts of the parabola representing the quadratic function.

• Calculus: Factorisation is crucial in finding derivatives and integrals of polynomial functions.

• Real-world Applications: Quadratic equations model various real-world phenomena, including projectile motion, area calculations, and optimization problems. Factorisation aids in solving these problems.

Frequently Asked Questions (FAQ)

Q1: What if the coefficient of x² is not 1?

A1: If the coefficient of x² is not 1, the factorisation becomes slightly more complex. Methods like completing the square or using the quadratic formula remain effective. Alternatively, you can use techniques like grouping or the AC method (for ax² + bx + c), where you find two numbers that add up to 'b' and multiply to 'a*c'.

Q2: Can all quadratic expressions be factorised?

A2: Not all quadratic expressions can be factorised using only integers. Some may require the use of irrational numbers or complex numbers. The discriminant (b² - 4ac) helps determine the nature of the roots and thus the possibility of integer factorisation. If the discriminant is a perfect square, integer factorisation is possible.

Q3: What if I get a different answer?

A3: Double-check your calculations. Ensure you correctly identified the factors of 'c' that add up to 'b'. Carefully expand your factorised form to verify if it matches the original expression. Remember, the order of factors doesn't matter (e.g., (x+2)(x+8) is the same as (x+8)(x+2)).

Q4: Are there any online tools to help with factorisation?

A4: While numerous online tools can factorise quadratic expressions, understanding the underlying methods is crucial for deeper comprehension and problem-solving. These tools should be used to verify your answers, not replace the learning process.

Conclusion

Factorising the quadratic expression x² + 10x + 16 is a straightforward process once you understand the underlying principles. We've explored three different methods – factorising by inspection, completing the square, and using the quadratic formula – each offering a unique approach to achieving the same result: (x + 2)(x + 8). Mastering these methods empowers you to tackle more complex quadratic expressions and opens doors to a deeper understanding of algebra and its many applications. Remember to practice regularly to build your confidence and proficiency. Don't hesitate to revisit these steps and methods as needed; understanding quadratic factorisation is a cornerstone of mathematical fluency.