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1. Learning Objectives
- Solve equations involving absolute values.
- Solve quadratic equations using factorization and the quadratic formula.
- Understand the discriminant ($b^2 - 4ac$) and the nature of roots.
- Solve basic exponential and radical equations.
2. Quick Notes
Absolute Value: If $|x| = a$, then $x = a$ or $x = -a$.
Quadratic Formula: For $ax^2 + bx + c = 0$, use:
x = [-b ± √(b² - 4ac)] / 2a
Exponents: If $a^x = a^y$, then $x = y$.
Click the button below each question to reveal the explanation.
1. Solve for x: |x| = 12
The distance from zero is 12. So x can be 12 or -12. Ans: 12, -12
2. Solve: |x + 5| = 8
Case 1: x + 5 = 8 → x = 3. Case 2: x + 5 = -8 → x = -13. Ans: 3, -13
3. Solve: |2x| = 10
2x = 10 → x = 5; 2x = -10 → x = -5. Ans: 5, -5
4. Solve: |x - 1| = -2
An absolute value represents distance and can never be negative. Ans: No Solution
5. Solve for x: x² = 49
x = ±√49. Ans: 7, -7
6. Solve: x² - 16 = 0
(x-4)(x+4) = 0. Ans: 4, -4
7. Solve by factoring: x² - 5x + 6 = 0
(x-2)(x-3) = 0. So x=2 or x=3. Ans: 2, 3
8. Solve: x² + 4x + 4 = 0
(x+2)² = 0. Ans: -2
9. Solve: 2x² = 18
Divide by 2: x² = 9. So x = ±3. Ans: 3, -3
10. Find the discriminant of x² + x + 1 = 0
D = b² - 4ac = 1² - 4(1)(1) = -3. Ans: -3
11. Nature of roots if D = 0?
When the discriminant is zero, the roots are real and equal. Ans: One Real Root
12. Solve: 2^x = 8
2^x = 2^3, so x = 3. Ans: 3
13. Solve: 5^x = 25
5^x = 5^2, so x = 2. Ans: 2
14. Solve: 3^(x-1) = 9
3^(x-1) = 3^2 → x-1 = 2 → x = 3. Ans: 3
15. Solve for x: √x = 5
Square both sides: x = 5² = 25. Ans: 25
16. Solve: √(x + 2) = 4
Square sides: x + 2 = 16 → x = 14. Ans: 14
17. Solve: |x/3| = 2
x/3 = 2 or x/3 = -2. Ans: 6, -6
18. Solve: x² - x - 12 = 0
(x-4)(x+3) = 0. Ans: 4, -3
19. Solve: 10^x = 1000
10^x = 10^3, so x = 3. Ans: 3
20. Nature of roots if D < 0?
A negative discriminant means no real number exists that satisfies the equation. Ans: No Real Roots