Algebra: Solving Linear and Quadratic Equations
What Is Algebra?
Algebra is a branch of Mathematics that deals with symbols and the rules for manipulating these symbols. It is a unifying thread of almost all of Mathematics and includes everything from solving simple equations to studying abstractions like groups, rings, and fields.
Linear Equations
Standard Form of a Linear Equation
A linear equation in one variable can be written in the form:
![\[ ax + b = 0 \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-879a438e9824ed064a59dd76da6a02aa_l3.png "Rendered by QuickLaTeX.com")
Where 
and 
are constants and 
is the variable.
Solving Linear Equations
To solve for , isolate on one side of the equation:
![\[ x = \frac{-b}{a} \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-ccd2abccc9783c3eb99bb43a92cca81b_l3.png "Rendered by QuickLaTeX.com")
Example: Solve 
.
Subtract 5 from both sides:
![\[ 3x = 6 \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-bd34ec3b49841dd4726c8f20e4228dce_l3.png "Rendered by QuickLaTeX.com")
Now divide by 3:
![\[ x = 2 \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-e3cdc7608b87d0b2337cb8eb41c876bb_l3.png "Rendered by QuickLaTeX.com")
Quadratic Equations
Standard Form of a Quadratic Equation
A quadratic equation in one variable can be written as:
![\[ ax^2 + bx + c = 0 \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-a980fbf483de1cbc18cd7034d985d70b_l3.png "Rendered by QuickLaTeX.com")
Where , , and 
are constants and 
.
Solving Quadratic Equations
- Factoring Method
For simple quadratics, you can factor the equation:
![\[ x^2 - 5x + 6 = 0 \quad \Rightarrow \quad (x - 2)(x - 3) = 0 \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-53df2ab52a301509bd00093a0715126a_l3.png "Rendered by QuickLaTeX.com")
So, the solutions are:
![\[ x = 2 \quad \text{or} \quad x = 3 \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-3a9a1cab763ca94f566c90f9d655e781_l3.png "Rendered by QuickLaTeX.com")
- Quadratic Formula
For more complex quadratics, use the quadratic formula:
![\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-406607c8b9fb91e3b47cc51fb4eb1348_l3.png "Rendered by QuickLaTeX.com")
Example: Solve 
.
![\[ x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(-5)}}{2(1)} = \frac{4 \pm \sqrt{16 + 20}}{2} = \frac{4 \pm \sqrt{36}}{2} \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-582334f169ddf0ca524a8a613c0de109_l3.png "Rendered by QuickLaTeX.com")
![\[ x = 5 \quad \text{or} \quad x = -1 \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-29ab97c0f68a886cbfdea74c0106a969_l3.png "Rendered by QuickLaTeX.com")
Applications of Algebra
Linear Equations in Business
Linear equations are used in financial calculations, such as determining profits, costs, and revenue.
Quadratic Equations in Science
Quadratic equations are used in Science, especially in problems related to motion, such as projectile motion.
Example Problem
Problem: Solve the quadratic equation 
.
- Solution Using Quadratic Formula:
![\[ x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(2)(4)}}{2(2)} \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-41aa032fa38ce4677a78b8ce3475a017_l3.png "Rendered by QuickLaTeX.com")
![\[ x = \frac{6 \pm \sqrt{36 - 32}}{4} = \frac{6 \pm \sqrt{4}}{4} = \frac{6 \pm 2}{4} \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-2b42e4564051fedf310c34fab7d215cc_l3.png "Rendered by QuickLaTeX.com")
Thus, 
or 
.
Common Mistakes in Algebra
- Incorrect Factoring: Always double-check factoring to avoid incorrect solutions.
- Quadratic Formula Misuse: Be careful with signs and the discriminant inside the square root.
- Dividing by Zero: Ensure the coefficient
in quadratic equations.
Practice Questions
- Solve
. - Factor
. - Explain why the quadratic formula works for all quadratic equations.
.
.
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