Algebra: Complete Notes, Formulas, Examples, and Practice Questions

1. Linear Equations

A. What is a Linear Equation?

An equation of the form $a x + b = 0$ or $a x + b y + c = 0$ , where $a, b, c$ are constants and $x, y$ are variables.

B. How to Solve

Isolate the variable on one side.
For two variables, use substitution or elimination methods.

Example 1:
Solve for $x$ : $2 x + 22 = x + 1$
$2 x + 22 = x + 1$
$2 x - x = 1 - 22$
$x = - 21$ 2

Example 2:
Solve the system:
$2 x + y = 8$
$3 y = 4 + 4 x$
Rewrite: $4 x - 3 y = - 4$
Solve simultaneously:
$x = 2, y = 4$ 2

2. Quadratic Equations

A. What is a Quadratic Equation?

An equation of the form $a x^{2} + b x + c = 0$ , where $a \neq 0$ .

B. Quadratic Formula

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

C. Factorization

Express as $(x - p) (x - q) = 0$ , so $x = p$ or $x = q$ .

Example 3:
Solve for $x$ : $x^{2} - 8 x - 33 = 0$
$x^{2} - 11 x + 3 x - 33 = 0$
$(x - 11) (x + 3) = 0$
$x = 11$ or $x = - 3$ 2

Example 4:
Solve: $x^{2} + 5 x + 6 = 0$
$x = \frac{- 5 \pm \sqrt{25 - 24}}{2} = \frac{- 5 \pm 1}{2}$
$x = - 2, - 3$ 4

3. Algebraic Identities

A. Common Identities4 5 7

$(a + b)^{2} = a^{2} + 2 a b + b^{2}$
$(a - b)^{2} = a^{2} - 2 a b + b^{2}$
$(a + b) (a - b) = a^{2} - b^{2}$
$(a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 a b + 2 b c + 2 c a$
$(a + b)^{3} = a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}$
$(a - b)^{3} = a^{3} - 3 a^{2} b + 3 a b^{2} - b^{3}$
$a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$
$a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$

Example 5:
Expand $(2 x - 3 y)^{2}$
$(2 x)^{2} - 2 \times 2 x \times 3 y + (3 y)^{2} = 4 x^{2} - 12 x y + 9 y^{2}$ 4

Example 6:
Evaluate $297 \times 303$
Write as $(300 - 3) (300 + 3) = 300^{2} - 3^{2} = 90000 - 9 = 89991$ 4

4. Inequalities

A. What is an Inequality?

A mathematical statement showing the relationship between expressions using $>, <, \geq, \leq$ .

B. Solving Linear Inequalities

Similar to equations, but when multiplying/dividing by a negative, reverse the inequality sign.

Example 7:
Solve: $2 x - 5 < 9$
$2 x < 14$
$x < 7$

5. Practice Questions (with answers for first 20)

Questions 1–20 (with answers)

Solve for $x$ : $5 x - 7 = 18$
Ans: $x = 5$
Solve for $x$ : $3 x + 2 = 8$
Ans: $x = 2$
Solve: $x^{2} - 4 x - 21 = 0$
Ans: $x = 7, - 3$
Expand: $(a + 2 b)^{2}$
Ans: $a^{2} + 4 a b + 4 b^{2}$
Factorize: $x^{2} + 7 x + 12$
Ans: $(x + 3) (x + 4)$
Solve: $2 x + 3 y = 12, x - y = 1$
Ans: $x = 3, y = 2$
Solve for $x$ : $x^{2} - 5 x + 6 = 0$
Ans: $x = 2, 3$
Expand: $(a - b)^{3}$
Ans: $a^{3} - 3 a^{2} b + 3 a b^{2} - b^{3}$
Solve: $4 x - 7 < 9$
Ans: $x < 4$
Solve: $x^{2} + 2 x - 15 = 0$
Ans: $x = 3, - 5$
Factorize: $x^{2} - 9$
Ans: $(x + 3) (x - 3)$
Expand: $(2 a + 3 b)^{2}$
Ans: $4 a^{2} + 12 a b + 9 b^{2}$
Solve: $x^{2} - 6 x + 9 = 0$
Ans: $x = 3$ (double root)
Solve: $2 x - 3 \leq 7$
Ans: $x \leq 5$
Factorize: $x^{2} - 2 x - 15$
Ans: $(x - 5) (x + 3)$
Expand: $(a + b) (a - b)$
Ans: $a^{2} - b^{2}$
Solve: $x^{2} + 8 x + 16 = 0$
Ans: $x = - 4$ (double root)
Solve: $x - 3 > 2$
Ans: $x > 5$
Expand: $(a + b + c)^{2}$
Ans: $a^{2} + b^{2} + c^{2} + 2 a b + 2 b c + 2 c a$
Solve for $y$ : $3 y - 7 = 2 y + 8$
Ans: $y = 15$

Questions 21–100 (Practice Only)

Solve for $x$ : $7 x + 5 = 26$
Solve for $x$ : $4 x - 3 = 13$
Solve: $x^{2} + 10 x + 24 = 0$
Expand: $(a - 2 b)^{2}$
Factorize: $x^{2} - 13 x + 36$
Solve: $3 x + 4 y = 18, x - y = 2$
Solve for $x$ : $x^{2} - 9 x + 18 = 0$
Expand: $(a + b)^{3}$
Solve: $5 x - 2 < 13$
Solve: $x^{2} - 2 x - 8 = 0$
Factorize: $x^{2} - 16$
Expand: $(3 a - 2 b)^{2}$
Solve: $x^{2} - 4 x + 4 = 0$
Solve: $4 x + 1 \geq 9$
Factorize: $x^{2} + 5 x + 6$
Expand: $(a - b)^{2}$
Solve: $x^{2} + 6 x + 9 = 0$
Solve: $2 x + 5 > 11$
Expand: $(a + 2 b + 3 c)^{2}$
Solve for $y$ : $2 y - 5 = 3 y + 4$
Algebra Practice Questions
Linear Equations (Single Variable & Systems)
1. Solve for $x$ : $2 x + 7 = 19$
2. Solve for $x$ : $5 x - 3 = 2 x + 12$
3. Solve for $x$ : $\frac{x}{3} + 4 = 10$
4. Solve for $x$ : $7 x + 2 = 4 x + 20$
5. Solve: $3 x - 2 = 2 x + 5$
6. If $2 x + 3 = 11$ , find $x$ .
7. If $4 x - 5 = 3 x + 8$ , find $x$ .
8. Solve for $x$ : $x - 7 = 2 x + 8$
9. Solve for $x$ : $9 x + 6 = 3 x + 24$
10. If $5 x + 2 = 3 x + 10$ , find $x$ .
11. Solve: $2 x - 3 y = 7$ , $x + y = 5$
12. Solve: $x + 2 y = 10$ , $2 x - y = 5$
13. Solve: $3 x + 4 y = 20$ , $2 x - y = 3$
14. If $x + y = 8$ and $x - y = 2$ , find $x$ and $y$ .
15. Solve: $2 x + 3 y = 12$ , $x - y = 1$
16. Solve: $x + y + z = 6$ , $x - y + z = 2$ , $x + y - z = 4$
17. Solve: $2 x - y = 5$ , $x + y = 7$
18. Solve: $x - 2 y = 4$ , $3 x + y = 11$
19. Solve: $x + 2 y = 7$ , $2 x - y = 3$
20. Solve: $x + y = 10$ , $x - y = 4$
Quadratic Equations
1. Solve: $x^{2} - 5 x + 6 = 0$
2. Solve: $x^{2} + 7 x + 12 = 0$
3. Solve: $x^{2} - 4 x - 21 = 0$
4. Solve: $2 x^{2} - 8 x + 6 = 0$
5. Solve: $x^{2} + 2 x - 15 = 0$
6. Solve: $3 x^{2} - 12 x + 9 = 0$
7. Solve: $x^{2} - 9 = 0$
8. Solve: $x^{2} + 6 x + 9 = 0$
9. Solve: $2 x^{2} - 5 x + 2 = 0$
10. Solve: $x^{2} - 7 x + 12 = 0$
11. Solve: $x^{2} - 2 x - 8 = 0$
12. Solve: $x^{2} + x - 6 = 0$
13. Solve: $x^{2} - 16 = 0$
14. Solve: $x^{2} + 4 x + 4 = 0$
15. Solve: $x^{2} - 6 x + 9 = 0$
16. Solve: $4 x^{2} - 12 x + 9 = 0$
17. Solve: $x^{2} - 10 x + 25 = 0$
18. Solve: $x^{2} + 5 x + 6 = 0$
19. Solve: $x^{2} - 3 x - 10 = 0$
20. Solve: $x^{2} - 49 = 0$
Systems Involving Linear & Quadratic Equations
1. Solve: $y = x^{2} - x - 6$ , $y = x - 3$ 8
2. Solve: $y = x^{2} + 2 x + 1$ , $y = 2 x + 3$
3. Solve: $x^{2} - 4 x + 3 = y$ , $x + y = 7$
4. Solve: $y = x^{2} - 5 x + 6$ , $y = 2 x - 1$
5. Solve: $x^{2} + y^{2} = 25$ , $x + y = 7$
6. Solve: $x^{2} - y = 3$ , $x + y = 5$
7. Solve: $x^{2} + y^{2} = 13$ , $x - y = 1$
8. Solve: $y = x^{2} - 2 x$ , $y = 3 x - 4$
9. Solve: $x^{2} + y^{2} = 10$ , $x + y = 4$
10. Solve: $x^{2} - y = 2$ , $x + y = 6$
Algebraic Identities and Expansions
1. Expand: $(a + b)^{2}$
2. Expand: $(a - b)^{2}$
3. Expand: $(a + b) (a - b)$
4. Expand: $(x + 2)^{2}$
5. Expand: $(3 x - 4)^{2}$
6. Expand: $(2 y + 5)^{2}$
7. Expand: $(a + b + c)^{2}$
8. Expand: $(x - 1)^{3}$
9. Expand: $(a + b)^{3}$
10. Expand: $(a - b)^{3}$
11. Factorize: $x^{2} - 9$
12. Factorize: $x^{2} + 6 x + 9$
13. Factorize: $x^{2} - 2 x - 8$
14. Factorize: $x^{2} - 5 x + 6$
15. Factorize: $x^{2} - 16$
16. Factorize: $x^{2} + 4 x + 4$
17. Factorize: $x^{2} - 7 x + 10$
18. Factorize: $x^{2} + 2 x - 15$
19. Factorize: $x^{2} - 1$
20. Factorize: $x^{2} - 6 x + 9$
Inequalities
1. Solve: $2 x + 5 > 11$
2. Solve: $3 x - 7 < 8$
3. Solve: $5 x + 2 \geq 17$
4. Solve: $4 x - 9 \leq 15$
5. Solve: $x - 3 > 2$
6. Solve: $2 x + 1 < 7$
7. Solve: $x + 4 \leq 10$
8. Solve: $3 x - 2 > 7$
9. Solve: $5 x + 3 < 23$
10. Solve: $2 x - 5 \geq 9$
11. Solve: $x^{2} - 4 x + 3 > 0$
12. Solve: $x^{2} - 9 < 0$
13. Solve: $x^{2} - 5 x + 6 \leq 0$
14. Solve: $x^{2} + 2 x - 8 \geq 0$
15. Solve: $x^{2} - 1 < 0$
16. Solve: $x^{2} + 4 x + 3 > 0$
17. Solve: $x^{2} - 2 x - 15 \leq 0$
18. Solve: $x^{2} - 7 x + 10 > 0$
19. Solve: $x^{2} + x - 12 < 0$
20. Solve: $x^{2} - 6 x + 8 \geq 0$
Mixed Algebra Applications
1. If $x + y = 10$ and $x y = 21$ , find $x^{2} + y^{2}$ .
2. If $a - b = 3$ and $a b = 10$ , find $a^{2} + b^{2}$ .
3. If $x^{2} + y^{2} = 25$ and $x y = 12$ , find $(x + y)^{2}$ .
4. If $x + y = 7$ and $x - y = 3$ , find $x y$ .
5. If $a + b = 5$ and $a - b = 1$ , find $a^{2} + b^{2}$ .
6. If $x^{2} - y^{2} = 15$ and $x - y = 3$ , find $x + y$ .
7. If $x^{2} + y^{2} = 20$ and $x + y = 6$ , find $x y$ .
8. If $a^{2} + b^{2} = 29$ and $a b = 10$ , find $(a + b)^{2}$ .
9. If $x^{2} + 2 x + 1 = 0$ , solve for $x$ .
10. If $x^{2} - 2 x + 1 = 0$ , solve for $x$ .
For more practice and detailed solutions, refer to sources like 3 4 5 6 78.
These questions comprehensively cover linear and quadratic equations, identities, and inequalities, as seen in competitive exams

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