Factorise 4x 2 4x 3

Factorising 4x² + 4x - 3: A Comprehensive Guide

This article provides a comprehensive guide on how to factorise the quadratic expression 4x² + 4x - 3. We'll explore various methods, delve into the underlying mathematical principles, and address frequently asked questions. Understanding quadratic factorisation is crucial for solving many algebraic problems, from finding roots of equations to simplifying complex expressions. This guide will equip you with the skills and knowledge to confidently tackle similar problems.

1. Introduction to Quadratic Expressions and Factorisation

A quadratic expression is a polynomial of degree two, meaning the highest power of the variable (in this case, 'x') is 2. It generally takes the form ax² + bx + c, where a, b, and c are constants. Factorisation involves breaking down a quadratic expression into a product of simpler expressions, usually two linear factors. This process is the reverse of expanding brackets.

Our target expression is 4x² + 4x - 3. We aim to find two linear expressions whose product equals this quadratic. This will allow us to easily solve equations where this expression is set to zero, find its roots, and simplify more complex algebraic manipulations.

2. Method 1: Finding Factors by Inspection (Trial and Error)

This method involves systematically trying different pairs of factors of the coefficient of x² (4) and the constant term (-3). It relies on intuition and familiarity with number patterns.

• Step 1: Identify factors of the coefficient of x² (4): The pairs are (1, 4) and (2, 2).

• Step 2: Identify factors of the constant term (-3): The pairs are (1, -3) and (-1, 3).

• Step 3: Test different combinations: We need to find a combination that, when multiplied and added, gives the coefficient of x (4). Let's try different combinations of these factors:

  • (1x + 1)(4x - 3): Expanding this gives 4x² - 3x + 4x - 3 = 4x² + x - 3. This is not correct.
  • (1x - 1)(4x + 3): Expanding this gives 4x² + 3x - 4x - 3 = 4x² - x - 3. This is not correct.
  • (1x + 3)(4x - 1): Expanding this gives 4x² - x + 12x - 3 = 4x² + 11x - 3. This is not correct.
  • (1x - 3)(4x + 1): Expanding this gives 4x² + x - 12x - 3 = 4x² - 11x - 3. This is not correct.
  • (2x + 3)(2x - 1): Expanding this gives 4x² - 2x + 6x - 3 = 4x² + 4x - 3. This is correct!

Therefore, the factorised form of 4x² + 4x - 3 is (2x + 3)(2x - 1).

3. Method 2: The Quadratic Formula

The quadratic formula provides a more systematic approach, especially for more complex quadratic expressions. It's a reliable method that always gives the roots (solutions) of a quadratic equation of the form ax² + bx + c = 0. The roots are then used to determine the factors.

The quadratic formula is: x = [-b ± √(b² - 4ac)] / 2a

For our expression 4x² + 4x - 3 = 0, we have a = 4, b = 4, and c = -3.

• Step 1: Substitute the values into the quadratic formula:

  x = [-4 ± √(4² - 4 * 4 * -3)] / (2 * 4) x = [-4 ± √(16 + 48)] / 8 x = [-4 ± √64] / 8 x = [-4 ± 8] / 8

• Step 2: Solve for the two roots:

  x₁ = (-4 + 8) / 8 = 4 / 8 = 1/2 x₂ = (-4 - 8) / 8 = -12 / 8 = -3/2

• Step 3: Convert roots to factors:

  If x₁ = 1/2, then (2x - 1) = 0 is a factor. If x₂ = -3/2, then (2x + 3) = 0 is a factor.

Therefore, the factorised form is (2x + 3)(2x - 1).

4. Method 3: Completing the Square

Completing the square is a technique used to rewrite a quadratic expression in the form (x + p)² + q. While it might seem less straightforward for this particular example, it's a valuable method to understand, especially for deriving the quadratic formula and understanding the geometry of parabolas.

• Step 1: Factor out the coefficient of x² from the x² and x terms:

  4(x² + x) - 3

• Step 2: Complete the square for the expression inside the brackets: To complete the square for x² + x, we take half of the coefficient of x (which is 1/2) and square it (1/4). We add and subtract this value inside the brackets:

  4(x² + x + 1/4 - 1/4) - 3

• Step 3: Rewrite as a perfect square:

  4[(x + 1/2)² - 1/4] - 3

• Step 4: Expand and simplify:

  4(x + 1/2)² - 1 - 3 4(x + 1/2)² - 4

• Step 5: Factor out a common factor:

  4[(x + 1/2)² - 1]

• Step 6: Use the difference of squares: (a² - b²) = (a + b)(a - b)

  4[(x + 1/2 + 1)(x + 1/2 - 1)] 4[(x + 3/2)(x - 1/2)] (2x + 3)(2x - 1)

Therefore, the factorised form is again (2x + 3)(2x - 1).

5. Explanation of the Mathematical Principles

The success of all these methods hinges on the distributive property (also known as the FOIL method – First, Outer, Inner, Last) of multiplication. When we expand (2x + 3)(2x - 1), we get:

• First: (2x)(2x) = 4x²
• Outer: (2x)(-1) = -2x
• Inner: (3)(2x) = 6x
• Last: (3)(-1) = -3

Adding these terms together gives 4x² + 4x - 3, confirming that (2x + 3)(2x - 1) is the correct factorisation. The quadratic formula is derived by completing the square on the general quadratic equation ax² + bx + c = 0.

6. Frequently Asked Questions (FAQs)

• Q: What if the quadratic expression cannot be factorised easily?

  A: If you cannot find integer factors by inspection or if the discriminant (b² - 4ac) in the quadratic formula is not a perfect square, the quadratic expression might have irrational or complex roots. In such cases, the quadratic formula will still provide the solutions, although the factors might be more complex.

• Q: Why is factorisation important?

  A: Factorisation is crucial for solving quadratic equations, simplifying algebraic expressions, finding the roots of a quadratic, sketching its graph (parabola), and performing various other algebraic manipulations.

• Q: Can I use a calculator or software to factorise quadratics?

  A: While calculators and software can assist with factorisation, understanding the underlying methods is essential for developing a strong foundation in algebra and problem-solving. These tools should be used to verify your work, not replace the learning process.

• Q: What if the leading coefficient (a) is negative?

  A: You can factor out a -1 from the entire expression before proceeding with any of the above methods. This simplifies the factorisation process.

7. Conclusion

Factorising quadratic expressions like 4x² + 4x - 3 is a fundamental skill in algebra. This article has explored three different methods – inspection, the quadratic formula, and completing the square – to achieve this factorisation, all resulting in the same solution: (2x + 3)(2x - 1). Understanding these methods will not only allow you to solve this specific problem but will also equip you with the tools to confidently tackle a wide range of similar problems in algebra and beyond. Remember to practice regularly to enhance your proficiency and understanding. The more you practice, the faster and more intuitively you'll be able to factorise quadratic expressions. Don't be afraid to try different methods and choose the one that feels most comfortable and efficient for you.