31 Quadratic Equation

Here are 31 examples of quadratic equations in standard form (ax² + bx + c = 0):

Simple Examples

1. x² + 5x + 6 = 0
2. x² - 9x + 20 = 0
3. 2x² + 7x + 3 = 0
4. 3x² - 2x - 1 = 0
5. x² - 4 = 0
6. x² + 9 = 0
7. 4x² - 12x + 9 = 0

Variations in Coefficients

8. -x² + 2x - 5 = 0
9. 5x² - x + 2 = 0
10. 1/2 x² + 3x - 4 = 0
11. -3x² - 10x + 1 = 0
12. x² + √2 x - 3 = 0

Applications and Word Problems (Converted to Equations)

13. x² - 100 = 0 (Finding the side length of a square with area 100)
14. h = -16t² + 64t + 80 (Projectile motion where h is height, t is time)
15. x(x + 7) = 120 (Finding dimensions of a rectangle with area 120)
16. -2x² + 500x - 10000 = 0 (Modeling profit where x is price)

More Complex Examples

17. 2x² - 5x + √3 = 0
18. x² + (1 + √5)x + √5 = 0
19. 3x² - (2 - √2)x + 1 = 0
20. x² - 3ix - 2 = 0 (Involving imaginary numbers)

Equations Reducible to Quadratic Form

These may not look like quadratics at first but can be transformed:

21. x⁴ - 5x² + 4 = 0 (Let y = x², then solve for y)
22. x - 5√x + 6 = 0 (Let y = √x, then solve for y)
23. 1/x + 1/(x+2) = 1 (Multiply through by a common denominator)
24. (x + 2)² - 3(x + 2) - 4 = 0 (Expand and rearrange)

Final Five!

25. x² + 4x - 1 = 0
26. 2x² - 6x + 1 = 0
27. x² - 8x + 16 = 0
28. x² - x - 6= 0
29. 5x² + 2x - 3 = 0
30. x² + 6x + 5 = 0
31. 3x² - x - 2 = 0

Answers:

Here are the answers to the 31 quadratic equations. Remember, most quadratic equations have two solutions.

1. x = -2, x = -3
2. x = 4, x = 5
3. x = -1/2, x = -3
4. x = 1, x = -1/3
5. x = 2, x = -2
6. x = 3i, x = -3i (imaginary solutions)
7. x = 3/2 (double root)
8. x = 1 ± √6
9. x = (1 ± √79i) / 10 (imaginary solutions)
10. x = -3, x = 4/3
11. x = (-5 ± √113) / -3
12. x = (-√2 ± √6) / 2
13. x = 10, x = -10
14. (Depends on the specific problem setup)
15. (Depends on the specific problem setup)
16. (Depends on the specific problem setup)
17. x = (5 ± √17) / 4
18. x = (-1 - √5 ± √(2 - 2√5)) / 2
19. x = (2 - √2 ± √(-2 - 2√2)) / 6
20. x = i, x = 2i (imaginary solutions)
21. x = 2, x = -2, x = 1, x = -1
22. x = 9, x = 4
23. x = 2, x = -3
24. x =3, x = -2
25. x = (-4 ± √20) /2 = -2 ± √5
26. x = (3 ± √7) / 2
27. x = 4 (double root)
28. x = 3, x = -2
29. x = -1, x = 3/5
30. x = -5, x = -1
31. x = 1, x = -2/3

Important Note: Equations 14-16 would require more information about the specific word problem to provide numerical answers.

Here's a breakdown of key concepts, methods, and examples related to quadratic equations:

What is a Quadratic Equation?

A quadratic equation is a mathematical equation of the second degree, meaning the highest power of the variable is 2. Its standard form looks like this:

• ax² + bx + c = 0
  • a, b, and c are coefficients (numbers) where 'a' cannot be zero.
  • x is the variable

Solving Quadratic Equations

There are several methods to solve quadratic equations:

1. Factoring: Try to factor the quadratic expression into two linear expressions. If successful, set each factor equal to zero and solve for x.

   • Example: x² + 5x + 6 = 0 factors into (x + 3)(x + 2) = 0. Therefore x = -3 or x= -2

2. Quadratic Formula: This formula provides solutions for any quadratic equation:

   • x = (-b ± √(b² - 4ac)) / 2a

3. Completing the Square: This involves manipulating the equation to create a perfect square trinomial and then solving by taking the square root of both sides.

Nature of Roots (Solutions)

The part of the quadratic formula under the radical (b² - 4ac) is called the discriminant and determines the nature of the roots:

• b² - 4ac > 0: Two distinct real roots.
• b² - 4ac = 0: One real root (a double root).
• b² - 4ac < 0: Two complex roots (involving imaginary numbers).

Examples

1. x² - 7x + 10 = 0

   • Factoring: (x - 5)(x - 2) = 0 => x = 5 or x = 2

2. 2x² + 3x - 1 = 0

   • Quadratic Formula: x = (-3 ± √(3² - 42(-1))) / 2*2 => x = -1.5 or x = 0.5

Applications

Quadratic equations have wide applications in physics, engineering, finance, and other fields. Examples include:

• Projectile motion
• Calculating areas and dimensions
• Optimization problems