Algebra 1.1

Algebra · 1.1  |  Core Knowledge Points

Chapter 1 · Algebra

Core Algebra —
Complete Knowledge Points

§ 6 Topics · Exponents · Radicals · Factoring · Equations · Inequalities · Systems

Contents

1. Laws of Exponents
2. Laws of Radicals
3. Factoring Polynomials
4. Solving Equations
5. Inequalities
6. Systems of Equations

01

Laws of Exponents

Let $a, b$ be real numbers, and $m, n$ be integers. The rules are:

Rule	Formula
Product rule	$a^m \cdot a^n = a^{m+n}$
Quotient rule	$\dfrac{a^m}{a^n} = a^{m-n}$
Power rule	$(a^m)^n = a^{mn}$
Product to a power	$(ab)^n = a^n b^n$
Quotient to a power	$\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$
Zero exponent	$a^0 = 1 \quad (a \neq 0)$
Negative exponent	$a^{-n} = \dfrac{1}{a^n}$

02

Laws of Radicals

A radical is just another way to write a fractional exponent:

$$\sqrt[n]{a} = a^{1/n} \qquad \sqrt[n]{a^m} = a^{m/n}$$

Rule	Formula
Product rule	$\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}$
Quotient rule	$\dfrac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\dfrac{a}{b}}$
Power rule	$\left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m}$
Simplifying	$\sqrt[n]{a^n} = a \quad (a \geq 0)$

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Important: $\sqrt{a^2} = |a|$, not simply $a$. For example, $\sqrt{(-3)^2} = \sqrt{9} = 3$, not $-3$.

Rationalizing the Denominator

Eliminate radicals from the denominator using these identities:

$$\frac{1}{\sqrt{a}} = \frac{\sqrt{a}}{a} \qquad \frac{1}{\sqrt{a} - \sqrt{b}} = \frac{\sqrt{a} + \sqrt{b}}{a - b}$$

03

Factoring Polynomials

Factoring means rewriting a polynomial as a product of simpler expressions.

3.1  Common Factor (GCF)

Always check this first:

$$6x^3 + 9x^2 = 3x^2(2x + 3)$$

3.2  Difference of Squares

$$a^2 - b^2 = (a+b)(a-b)$$

Example: $x^2 - 9 = (x+3)(x-3)$

3.3  Perfect Square Trinomials

$$a^2 + 2ab + b^2 = (a+b)^2$$ $$a^2 - 2ab + b^2 = (a-b)^2$$

3.4  Sum / Difference of Cubes

$$a^3 + b^3 = (a+b)(a^2 - ab + b^2)$$ $$a^3 - b^3 = (a-b)(a^2 + ab + b^2)$$

3.5  Factoring Trinomials $ax^2 + bx + c$

Find two numbers that multiply to $ac$ and add to $b$:

$$x^2 + 5x + 6 = (x+2)(x+3)$$

For $a \neq 1$, use the AC method or trial and error:

$$2x^2 + 7x + 3 = (2x + 1)(x + 3)$$

04

Solving Equations

4.1  Linear Equations

Equations of the form $ax + b = 0$. Just isolate $x$:

$$3x - 6 = 0 \implies x = 2$$

4.2  Quadratic Equations   $ax^2 + bx + c = 0$

Three methods:

Method 1 — Factoring

$$x^2 - 5x + 6 = 0 \implies (x-2)(x-3) = 0 \implies x = 2 \text{ or } x = 3$$

Method 2 — Completing the Square

$$x^2 + 6x + 5 = 0$$ $$x^2 + 6x = -5$$ $$(x+3)^2 = 4$$ $$x = -3 \pm 2$$

Method 3 — Quadratic Formula (works for any quadratic)

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

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The expression $\Delta = b^2 - 4ac$ is called the discriminant:

Condition	Number of Real Roots
$\Delta > 0$	Two distinct real roots
$\Delta = 0$	One repeated real root
$\Delta < 0$	No real roots (two complex roots)

Want to see where this formula comes from? Proof of Quadratic Formula

4.3  Rational Equations

Equations with variables in the denominator. Multiply both sides by the LCD, then check for extraneous solutions (values that make the denominator zero):

$$\frac{1}{x} + \frac{1}{x+1} = \frac{2}{3}$$

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Always verify your answers back in the original equation.

05

Inequalities

5.1  Linear Inequalities

Solve like an equation, but flip the inequality sign when multiplying or dividing by a negative number:

$$-2x > 6 \implies x < -3$$

5.2  Quadratic Inequalities

To solve $ax^2 + bx + c > 0$:

1. Find the roots (solve $= 0$)
2. Draw a number line and test each interval
3. Pick the intervals where the inequality holds

Example: $x^2 - x - 6 > 0$

Roots: $x = 3$ and $x = -2$  →  Solution: $x < -2$ or $x > 3$

5.3  Absolute Value Inequalities

Form	Meaning	Solution
$|x| < a$	Distance from 0 is less than $a$	$-a < x < a$
$|x| > a$	Distance from 0 is greater than $a$	$x < -a$ or $x > a$

06

Systems of Equations

A system has two or more equations with the same variables. You find the values that satisfy all equations simultaneously.

6.1  Substitution Method

Solve one equation for one variable, then substitute into the other:

$$\begin{cases} y = 2x + 1 \\ 3x + y = 11 \end{cases} \implies 3x + (2x+1) = 11 \implies x = 2,\ y = 5$$

6.2  Elimination Method

Add or subtract equations to cancel one variable:

$$\begin{cases} 2x + 3y = 12 \\ 2x - y = 4 \end{cases} \implies \text{subtract} \implies 4y = 8 \implies y = 2,\ x = 3$$

Types of Solutions

Case	Geometric Meaning	Solutions
Lines intersect	Different slopes	Exactly one solution
Lines are parallel	Same slope, different intercepts	No solution
Lines are the same	Identical equations	Infinitely many solutions