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:
- Final answer: \( -x^2 + 7x \).
\[
3x + 4x = 7x, \quad -2x^2 + x^2 = -x^2.
\]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 \( 3x + 6 \).
- Find the common factor: \( 3 \).
- Write as \( 3(x + 2) \).
Solving Linear Equations
Linear equations have the form \( ax + b = c \).
Steps to Solve:
- Isolate \( x \): Undo addition or subtraction.
- Divide by the coefficient of \( x \).
Example: Solve \( 3x + 2 = 11 \).
- Subtract 2: \( 3x = 9 \).
- Divide by 3: \( x = 3 \).
Solving Quadratic Equations
Quadratics have the form \( ax^2 + bx + c = 0 \).
Methods:
Factorisation:
- Example: Solve \( x^2 + 5x + 6 = 0 \).
- Factorise: \( (x + 2)(x + 3) = 0 \).
- Solutions: \( x = -2, x = -3 \).
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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