Master more advanced algebra with concepts such as brackets, fractions, simultaneous equations, and quadratic equations.
1. Brackets in Algebra
-
Single Bracket: Expand the terms inside:
[
4(x - 2) = 18 \quad \Rightarrow \quad 4x - 8 = 18 \quad \Rightarrow \quad x = 6.5
]
-
Two Brackets (FOIL Method):
Multiply terms in First, Outer, Inner, Last order:
[
(2x + 5)(x + 4) = 0
]
Expand:
[
2x^2 + 8x + 5x + 20 = 2x^2 + 13x + 20
]
2. Equations with Fractions??
- Use Cross-Multiplication to eliminate fractions:
[
\frac{2 + x}{3} = \frac{9 + x}{5}
]
Steps:
- Multiply through by denominators (3 and 5):
[
5(2 + x) = 3(9 + x)
]
- Simplify:
[
10 + 5x = 27 + 3x \quad \Rightarrow \quad 2x = 17 \quad \Rightarrow \quad x = 8.5
]
3. Simultaneous Equations
- Rule: The number of equations must equal the number of unknowns.
- Method:
- Solve one equation for one variable ((x = f(y))).
- Substitute into the second equation.
- Solve for (y), then back-substitute to find (x).
Example 1: Simple substitution
[
\begin{aligned}
2x = 6, \quad y = 4x + 5 \
x = 3, \quad y = 4(3) + 5 = 17
\end{aligned}
]
Example 2: When (x = f(y))
[
\begin{aligned}
x - y = 1, \quad 2x + 3y = 27 \
x = y + 1 \quad \Rightarrow \quad 2(y + 1) + 3y = 27 \quad \Rightarrow \quad y = 5, \, x = 6
\end{aligned}
]
4. Quadratic Equations
Form:
[
ax^2 + bx + c = 0
]
- Graph: Quadratic equations produce parabolas.
- Roots = (x)-intercepts (where (y = 0)).
Methods to Solve Quadratics
- Factorization:
- Break into two brackets:
[
x^2 + 9x + 20 = 0 \quad \Rightarrow \quad (x + 4)(x + 5) = 0
]
-
Roots:
[
x = -4, \, x = -5
]
-
Quadratic Formula:
- For any (ax^2 + bx + c = 0):
[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
]
-
Example: Solve (2x^2 + 5x - 3 = 0):
[
a = 2, \, b = 5, \, c = -3 \quad \Rightarrow \quad x = \frac{-5 \pm \sqrt{5^2 - 4(2)(-3)}}{2(2)}
]
-
Completing the Square:
- Rewrite (x^2 - 18x + 72 = 0) as:
[
(x - 9)^2 = 9
]
- Roots:
[
x = 9 \pm 3 \quad \Rightarrow \quad x = 12, \, x = 6
]
Summing it up
- Always perform the same operation on both sides of the equation.
- Master the FOIL method for expanding brackets.
- Use cross-multiplication to simplify equations with fractions.
- For quadratics, choose between factorization, formula, or completing the square based on complexity.
? Equations are like puzzles—use logic to solve step by step!