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
.- Combine like terms:
- Final answer:
.
![\[ 3x + 4x = 7x, \quad -2x^2 + x^2 = -x^2. \]](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNjciIGhlaWdodD0iMjEiIHZpZXdCb3g9IjAgMCAyNjcgMjEiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+ "Rendered by QuickLaTeX.com")
Expanding Brackets
Multiply each term inside the bracket by the term outside.
Example: Expand
.![\[ 2(x + 3) = 2x + 6. \]](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxNDEiIGhlaWdodD0iMTkiIHZpZXdCb3g9IjAgMCAxNDEgMTkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+ "Rendered by QuickLaTeX.com")
Factorising Expressions
Factorising reverses expansion.
Example: Factorise
.- Find the common factor:
. - Write as .
.![\[ 3x + 4x = 7x, \quad -2x^2 + x^2 = -x^2. \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-4206ff4ec51f89e8191989b5a3ace0d7_l3.png "Rendered by QuickLaTeX.com")
.
.![\[ 2(x + 3) = 2x + 6. \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-ea65237dc4580bf36f0c864260ae5f0a_l3.png "Rendered by QuickLaTeX.com")
.
.
.Solving Linear Equations
Linear equations have the form
.Steps to Solve:
- Isolate
: Undo addition or subtraction. - Divide by the coefficient of .
Example: Solve
.- Subtract 2:
. - Divide by 3:
.
- Isolate
.
: Undo addition or subtraction.
.
.
.Solving Quadratic Equations
Quadratics have the form
.Methods:
Factorisation:
- Example: Solve
. - Factorise: .
- Solutions:
.
- Example: Solve
Quadratic Formula:
![\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \]](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxNjMiIGhlaWdodD0iNDAiIHZpZXdCb3g9IjAgMCAxNjMgNDAiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+ "Rendered by QuickLaTeX.com")
.
.
.
.![\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-a9c6dbcb78488f7b8785ad1fc73732a6_l3.png "Rendered by QuickLaTeX.com")
Practice Question
Question: Solve 
.
Solution:
- Subtract 5:
. - Divide by 2:
.
.
.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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