Solving_Quadratic_Equations___Prime_Number_Formula

Solving Quadratic Equations / Prime Number Formula Project 1: An interesting method for solving quadratic equations came from India. I will provide you the necessary steps to solve the quadratic equation. Then I will have four examples that will show the formula in use. Step 1 – Move the constant term to the right side of the equation. Step 2 – Multiply each term in the equation by four times the coefficient of the x2 term. Step 3 – Square the coefficient of the original x term and add it to both sides of the equation. Step 4 – Take the square root of both sides. Step 5 – Set the left side of the equation equal to the positive square root of the number on the right side and solve for x. Step 6 – Set the left side of the equation equal to the negative square root of the number on the right side of the equation and solve for x. (a) First example: x2 – 2x – 13 = 0 (Step 1) x2 – 2x – 13 + 13 = 0 + 13 x2 – 2x = 13 (Step 2) 4x2 – 8x = 52 (Step 3) 4x2 – 8x + 4 = 52 + 4 4x2 – 8x + 4 = 56 (Step 4) 2x – 2 = + 7.48 (Step 5) (Step 6) 2x – 2 = 7.48 | 2x – 2 = - 7.48 + 2 + 2 | + 2 + 2 | 2x = 9.48 | 2x = - 5.48 2 2 | 2 2 | x = 4.74 | x = - 2.74 (b) Second example: 4x2 – 4x + 3 = 0 (Step 1) 4x2 – 4x + 3 - 3 = 0 - 3 4x2 – 4x = - 3 (Step 2) 64x2 – 64x = - 48 (Step 3) 64x2 – 64x + 16 = - 48 + 16 16(2x – 1)2 = - 32 (Step 4) 4(2x – 1)2 = + - 4[2]i (Step 5) (Step 6) 2x – 2 = [2]i | 2x – 2 = - [2]i + 2 + 2 | + 2 + 2 | 2x = [2]i | 2x = - [2]i 2 2 | 2 2 | x = (1 + ^[2]i)/2 | x = (1 - ^[2]i)/2 (c) Third example: x2 + 12x – 64 = 0 (Step 1) x2 + 12x – 64 + 64 = 0 + 64 x2 + 12x = 64 (Step 2) 4x2 + 48x = 256 (Step 3) 4x2 + 48x + 144 = 256 + 144 4x2 + 48x + 144 = 400 (Step 4) 2x + 12 = + 20 (Step 5) (Step 6) 2x + 12 = 20 | 2x + 12 = - 20 - 12 - 12 | - 12 - 12 | 2x = 8 | 2x = - 32 2 2 | 2 2 | x = 4 | x = - 16 (d) Forth example: 2x2 – 3x – 5 = 0 (Step 1) 2x2 – 3x – 5 + 5 = 0 + 5 2x2 – 3x = 5 (Step 2) 16x2 – 24x = 40 (Step 3) 16x2 – 24x + 9 = 40 + 9 16x2 – 24x + 9 = 49 (Step 4) 4x – 3 = + 7 (Step 5) (Step 6) 4x – 3 = 7 | 4x – 3 = - 7 + 3 + 3 | + 3 + 3 | 4x = 10 | 4x = - 4 4 4 | 4 4 | x = 2.5 | x = - 1 Project 2: What is a mathematic formula that yields prime numbers' Mathematicians have been searching for a formula that yields prime numbers. One such formula is x2 – x + 41. In the following examples (1-5), I will be using five numbers that will include a zero, two even and two odd numbers. I will also try to find a number for x (examples 6-8) that when substituted in the formula will yield a composite number. Example 1: x = 0 (zero). x2 – x + 41 02 – 0 + 41 = 41, (which is a prime number) Example 2: x = 2 (even #). x2 – x + 41 22 – 2 + 41 4 – 2 + 41 2 + 41 = 43, (which is a prime number) Example 3: x = 4 (even #). x2 – x + 41 42 – 4 + 41 16 – 4 + 41 12 + 41 = 53, (which is a prime number) Example 4: x = 5 (odd #). x2 – x + 41 52 – 5 + 41 25 – 5 + 41 20 + 41 = 61, (which is a prime number) Example 5: x = 9 (odd #). x2 – x + 41 92 – 9 + 41 81 – 9 + 41 72 + 41 = 113, (which is a prime number) Example 6: x = 41. x2 – x + 41 412 – 41 + 41 1681 – 41 + 41 1640 + 41 = 1681, (which is not prime, it has 41*41) Example 7: x = 82. x2 – x + 41 822 – 82 + 41 6724 – 82 + 41 6642 + 41 = 6683, (which is not prime, it has 41*163) Example 8: x = 123. x2 – x + 41 1232 – 123 + 41 15129 – 123 + 41 15006 + 41 = 15047, (which is not prime, it has 41*367) Conclusion In conclusion, Project 1 consisted in an interesting method for solving quadratic equations that came from India. The method required six steps to calculate the four problems. The formula used step by step will result in the correct calculations. In Project 2 we looked at the formula that mathematicians used to yield prime numbers. I found it interesting that when I used 41 and then doubled it getting 82 and then tripled 41 it still led to a composite number. This math I must admit was a bit challenging. It took a lot of figuring out how to work the quadratic equations but hopefully I have come up with the correct answers. References Bluman, A.G. (2005). Mathematics in Our World. Dubuque, IA: The McGraw-Hill Companies, Inc..