Mathematics EN 9th grade: First and second degree equations

1. First-degree equations

Form:

$ ax + b = 0 $

✔ Example:

$ 2x + 4 = 0 \Rightarrow x = -2 $

2. First-degree inequalities

✔ Example:

$ 2x + 1 > 5 \Rightarrow x > 2 $

3. Quadratic Equations

- General Form

$ ax^2 + bx + c = 0 $, com $ a \neq 0 $

- Zero-Product Law

$ ab = 0 \Rightarrow a = 0 \ \text{ou} \ b = 0 $

- Incomplete Equations

✔ Example:

$ x^2 - 9 = 0 \Rightarrow x = \pm 3 $

- Complete Equations

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

✔ Example:

$ x^2 - 5x + 6 = 0 \Rightarrow x = 2 \ \text{ou} \ x = 3 $

4. Literal Equations

Solving for a function of one variable

✔ Example:

$ ax + b = 0 \Rightarrow x = -\frac{b}{a} $

5. Systems of Equations

✔ Example:

$ \begin{cases} x + y = 5 \\ x - y = 1 \end{cases} \Rightarrow x = 3,\ y = 2 $

- Classification:

Possible and determined system (one solution)

Impossible system (no solution)

Indeterminate system (infinite solutions)

QUICK TIPS FOR THE EXAM/b>

✔ Similar terms → same literal part

✔ Always distribute correctly

✔ Notable cases → memorize formulas

✔ Zero product → separate into two equations

✔ Always check the solutions

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$Explore more about What are rational numbers? They are numbers that can be written in fraction form: $ \frac{a}{b}, \quad b \neq 0 $ Examples: $ \frac{3}{4}, -2, 0{,}5, -\frac{7}{3} $ valor absoluto The valor absoluto represents the distance to zero: $ |a| = a $ se $ a \geq 0 $ $ |a| = -a $ se $ a Examples: \( |5| = 5 $ $ |-5| = 5 $ Simétrico The Simétrico of a number is: $ -a $ Examples: $ 3 \rightarrow -3 $ $ -7 \rightarrow 7 $ Adição e subtração Addition with the same sign Add the values and keep the sign Examples: $ 3 + 5 = 8 $ $ -4 + (-6) = -10 $ Soma com sinais contrários Subtract the values and take the sign of the number with the greatest absolute valueNL# Examples: $ 7 + (-3) = 4 $ $ -8 + 5 = -3 $ Soma de números simétricos The result is always zero $ a + (-a) = 0 $ Examples: $ 5 + (-5) = 0 $ Subtração (diferença) It is transformed into the sum of the opposite $ a - b = a + (-b) $ Examples: $ 7 - 3 = 7 + (-3) = 4 $ $ 5 - (-2) = 5 + 2 = 7 $ Divisão de números racionais Main rule Multiply by the reciprocal $ \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} $ Inverso de um número racional Swap the numerator and denominator $ \text{Inverso de } \frac{a}{b} = \frac{b}{a}, \quad a \neq 0 $ Examples: $ \frac{3}{4} \rightarrow \frac{4}{3} $ $ 2 = \frac{2}{1} \rightarrow \frac{1}{2} $ Important: $ 0 \text{ não tem inverso} $ Examples de divisão $ \frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6} $ TIPS ✔ Same signs → add and keep the sign ✔ Different signs → subtract and take the sign of the number with the greatest absolute value ✔ Subtraction → convert to addition ✔ Division → multiply by the reciprocal ✔ Número + simétrico: $ a + (-a) = 0 $ ✔ Valor absoluto: $ |a| \geq 0 $ ..$
What are rational numbers?
They are numbers that can be written in fraction form:
$ \frac{a}{b}, \quad b \neq 0 $
Examples:
$ \frac{3}{4}, -2, 0{,}5, -\frac{7}{3} $
valor absoluto
The valor absoluto represents the distance to zero:
$ |a| = a $ se $ a \geq 0 $
$ |a| = -a $ se $ a < 0 $
Examples:
$ |5| = 5 $
$ |-5| = 5 $
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