Simplifying Algebraic Expressions and Solving Equations

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Simplifying Algebraic Expressions and Solving Equations

Introduction

Algebra is a cornerstone of GCSE Maths. From simplifying expressions to solving equations, mastering these skills is essential for exam success.

In this article, we’ll focus on:

Simplifying algebraic expressions.
Solving linear and quadratic equations.
Real-life applications of algebra.

Simplifying Algebraic Expressions
Combining Like Terms
Terms with the same variable and power can be added or subtracted.
Example: Simplify $3x + 4x - 2x^2 + x^2$ .
- - Combine like terms:
$3x + 4x = 7x, \quad -2x^2 + x^2 = -x^2.$
- Final answer: $-x^2 + 7x$ .
Expanding Brackets
Multiply each term inside the bracket by the term outside.
Example: Expand $2(x + 3)$ .
$2(x + 3) = 2x + 6.$
Factorising Expressions
Factorising reverses expansion.
Example: Factorise $3x + 6$ .
1. Find the common factor: $3$ .
2. Write as $3(x + 2)$ .

Solving Linear Equations
Linear equations have the form $ax + b = c$ .
Steps to Solve:
1. Isolate $x$ : Undo addition or subtraction.
2. Divide by the coefficient of $x$ .
Example: Solve $3x + 2 = 11$ .
1. Subtract 2: $3x = 9$ .
2. Divide by 3: $x = 3$ .

Solving Quadratic Equations
Quadratics have the form $ax^2 + bx + c = 0$ .
Methods:
1. Factorisation:
  - Example: Solve $x^2 + 5x + 6 = 0$ .
  - Factorise: $(x + 2)(x + 3) = 0$ .
  - Solutions: $x = -2, x = -3$ .
2. Quadratic Formula:
  $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.$

Practice Question

Question: Solve $2x + 5 = 15$ .

Solution:

Subtract 5: $2x = 10$ .
Divide by 2: $x = 5$ .

Conclusion

Simplifying expressions and solving equations are key algebraic skills for GCSE Maths. Practice consistently to strengthen your understanding and excel in exams.

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