How to generate the Almighty Quadratic Formula скачать в хорошем качестве

How to generate the Almighty Quadratic Formula

After watching this video, you would be able to generate the quadratic formula by solving the quadratic equation ax²+bc+c=0, using the method of completing the squares. Quadratic Equations A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (usually x) is two. The general form of a quadratic equation is: ax² + bx + c = 0 where a, b, and c are constants, and a ≠ 0. Methods for Solving Quadratic Equations 1. *Factoring*: Factor the quadratic expression into two binomial factors. 2. *Quadratic Formula*: Use the formula: x = (-b ± √(b² - 4ac)) / 2a 3. *Completing the Square*: Complete the square to rewrite the equation in vertex form. Example 1: Factoring Solve: x² + 5x + 6 = 0 Solution 1. Factor: (x + 3)(x + 2) = 0 2. Solve: x + 3 = 0 or x + 2 = 0 3. Solutions: x = -3 or x = -2 Example 2: Quadratic Formula Solve: 2x² - 7x - 3 = 0 Solution 1. Identify a, b, c: a = 2, b = -7, c = -3 2. Plug into the formula: x = (7 ± √((-7)² - 4(2)(-3))) / 2(2) 3. Simplify: x = (7 ± √(49 + 24)) / 4 4. Solutions: x = (7 + √73) / 4 or x = (7 - √73) / 4 Tips 1. *Check the discriminant*: b² - 4ac determines the nature of the solutions. 2. *Use the quadratic formula*: When factoring is difficult or impossible. Solving Quadratic Equations by Completing the Square Let's solve the quadratic equation: ax² + bx + c = 0 Step-by-Step Solution 1. *Divide by a*: x² + (b/a)x + (c/a) = 0 2. *Move the constant term*: x² + (b/a)x = -c/a 3. *Add (b/2a)² to both sides*: x² + (b/a)x + (b/2a)² = (b/2a)² - c/a 4. *Factor the left side*: (x + b/2a)² = (b² - 4ac) / 4a² 5. *Take the square root*: x + b/2a = ±√((b² - 4ac) / 4a²) 6. *Solve for x*: x = (-b ± √(b² - 4ac)) / 2a Example Solve: x² + 6x + 5 = 0 Solution 1. Move the constant term: x² + 6x = -5 2. Add (6/2)² = 9 to both sides: x² + 6x + 9 = 4 3. Factor the left side: (x + 3)² = 4 4. Take the square root: x + 3 = ±2 5. Solve for x: x = -3 ± 2 6. Solutions: x = -1 or x = -5 Tips 1. *Check the discriminant*: b² - 4ac determines the nature of the solutions. 2. *Use completing the square*: When factoring is difficult or impossible. The Quadratic Formula The quadratic formula is a powerful tool for solving quadratic equations of the form: ax² + bx + c = 0 The formula is: x = (-b ± √(b² - 4ac)) / 2a How to Use the Quadratic Formula 1. *Identify a, b, and c*: Determine the values of a, b, and c in the quadratic equation. 2. *Plug into the formula*: Substitute the values of a, b, and c into the quadratic formula. 3. *Simplify*: Simplify the expression under the square root and solve for x. Example Solve: 2x² - 7x - 3 = 0 Solution 1. Identify a, b, c: a = 2, b = -7, c = -3 2. Plug into the formula: x = (7 ± √((-7)² - 4(2)(-3))) / 2(2) 3. Simplify: x = (7 ± √(49 + 24)) / 4 4. Solutions: x = (7 + √73) / 4 or x = (7 - √73) / 4 Discriminant (b² - 4ac) *b² - 4ac greater than 0*: Two distinct real solutions *b² - 4ac = 0*: One repeated real solution *b² - 4ac less than 0*: No real solutions (complex solutions) #maths #algebra #mathshorts #education #learning #quadraticequation #mathtutorial Would you like to watch similar videos on solving quadratic equations or explore other similar topics? just drop a comment!