- Arithmetic β fluency with operations needed for equation solving
- Fractions β variables often appear in fractional equations
π€ Algebra Study Guide
Algebra is the language of mathematics. It lets you solve problems with unknowns. From calculating interest rates to engineering design, algebra is everywhere in modern technology and everyday decisions.
Think of algebra as a balance scale. The equals sign (=) means both sides must always balance. Whatever you do to one side, you must do to the other. This is why moving a term across the equals sign changes its sign.
Common mistake: Moving a term across the equals sign just reverses its sign.
Reality: This happens because you subtract the same value from both sides to maintain balance. x + 3 = 7 means x = 7 minus 3. Understanding the reason means you will never forget the rule.
1. Variables and Expressions
Algebra uses symbols (usually letters) to represent unknown values or quantities that can change.
- Variable: a letter representing an unknown value (e.g., \(x\), \(y\), \(n\))
- Constant: a fixed number that does not change (e.g., 5, β3)
- Coefficient: the number multiplied by a variable (e.g., in \(4x\), the coefficient is 4)
- Term: a single number, variable, or their product (e.g., \(3x^2\), \(-7\), \(5xy\))
- Expression: a combination of terms without an equals sign (e.g., \(2x + 5\))
- Equation: a statement that two expressions are equal, using "=" (e.g., \(2x + 5 = 13\))
- Like terms: terms with the same variable(s) raised to the same power (e.g., \(3x\) and \(-5x\))
\(4x^2 + 3x - 2x^2 + 7x - 5\)
\(= (4x^2 - 2x^2) + (3x + 7x) - 5\)
\(= 2x^2 + 10x - 5\)
2. Linear Equations
A linear equation has variables raised to the first power only. Solving means finding the value of the variable that makes the equation true. Use inverse operations to isolate the variable.
- Distribute if needed
- Combine like terms on each side
- Move variables to one side, constants to the other
- Divide both sides by the coefficient of the variable
Subtract 7: \(3x = 15\)
Divide by 3: \(x = 5\)
Distribute: \(8x - 12 = 3x + 7\)
Subtract \(3x\): \(5x - 12 = 7\)
Add 12: \(5x = 19\)
Divide by 5: \(x = \dfrac{19}{5} = 3.8\)
3. Linear Inequalities
Inequalities are solved like equations, with one critical rule: flip the inequality sign when multiplying or dividing by a negative number.
- \( < \) β less than
- \( > \) β greater than
- \( \leq \) β less than or equal to
- \( \geq \) β greater than or equal to
Subtract 6: \(-3x > -6\)
Divide by \(-3\) (flip sign!): \(x < 2\)
Graph: open circle at 2, shading to the left.
4. Systems of Linear Equations
A system of two equations with two unknowns has three possible outcomes: one solution (lines intersect), no solution (parallel lines), or infinitely many solutions (same line).
- Solve one equation for one variable
- Substitute that expression into the other equation
- Solve for the remaining variable
- Back-substitute to find the first variable
Substitute: \(3x + (2x-1) = 14\) β \(5x = 15\) β \(x = 3\)
Back-sub: \(y = 2(3) - 1 = 5\) Solution: (3, 5)
- Multiply equations (if needed) so that one variable has opposite coefficients
- Add the equations to eliminate that variable
- Solve for the remaining variable and back-substitute
Add: \(6x = 18\) β \(x = 3\)
Back-sub: \(2(3) + 3y = 12\) β \(3y = 6\) β \(y = 2\) Solution: (3, 2)
5. Quadratic Equations AP Exam
A quadratic equation has the form \(ax^2 + bx + c = 0\) where \(a \neq 0\). It can have 0, 1, or 2 real solutions.
Rewrite as a product of two binomials, then set each factor to zero.
Factor: \((x-3)(x+2) = 0\)
Solutions: \(x = 3\) or \(x = -2\)
For \(ax^2 + bx + c = 0\): \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
\(x = \dfrac{4 \pm \sqrt{16 + 48}}{4} = \dfrac{4 \pm \sqrt{64}}{4} = \dfrac{4 \pm 8}{4}\)
\(x = 3\) or \(x = -1\)
- Positive: two distinct real solutions
- Zero: one real solution (a "double root")
- Negative: no real solutions (two complex solutions)
6. Polynomials
A polynomial is a sum of terms with non-negative integer exponents. Operations with polynomials use the same rules as arithmetic.
First, Outer, Inner, Last
\((x+3)(x-5) = x^2 - 5x + 3x - 15 = x^2 - 2x - 15\)
- Perfect square: \((a+b)^2 = a^2 + 2ab + b^2\)
- Perfect square: \((a-b)^2 = a^2 - 2ab + b^2\)
- Difference of squares: \((a+b)(a-b) = a^2 - b^2\)
7. Factoring
Factoring reverses multiplication. Always check for a GCF first, then look for patterns.
- GCF: \(6x^2 + 9x = 3x(2x + 3)\)
- Difference of Squares: \(a^2 - b^2 = (a+b)(a-b)\)
Example: \(x^2 - 25 = (x+5)(x-5)\) - Perfect Square Trinomial: \(x^2 + 6x + 9 = (x+3)^2\)
- Trinomial (leading coeff. = 1): \(x^2 + bx + c\) β find two numbers that multiply to \(c\) and add to \(b\)
Example: \(x^2 + 5x + 6 = (x+2)(x+3)\) because \(2 \times 3 = 6\) and \(2+3=5\) - Trinomial (leading coeff. β 1): use the AC method or trial and error
Example: \(2x^2 + 7x + 3 = (2x+1)(x+3)\)
8. Practice Problems
- Simplify: \(5x^2 - 3x + 2x^2 + 8x - 4\)
- Solve: \(6x - 5 = 4x + 11\)
- Solve: \(-2x + 4 \leq 10\)
- Solve the system: \(x + y = 8\) and \(2x - y = 4\)
- Solve by factoring: \(x^2 - 7x + 10 = 0\)
- Use the quadratic formula: \(x^2 + 3x - 4 = 0\)
- Multiply: \((2x - 3)(3x + 4)\)
- Factor completely: \(4x^2 - 36\)
- When moving terms across the equals sign, flip the sign
- Quadratic formula: x = (βb Β± β(bΒ²β4ac)) / 2a
- Discriminant bΒ²β4ac: >0 two real roots, =0 one root, <0 no real roots
- Try factoring first β faster than the quadratic formula
Basic Q. Solve for x: 3x + 7 = 22
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Intermediate Q. A rectangle has perimeter 36 cm. Its length is twice its width. Find the dimensions.
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Advanced Q. Explain why the equation xΒ² + 1 = 0 has no real solutions, then find its complex solutions.
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Review this material at increasing intervals to commit it to long-term memory.