Exam Questions – Solved by the quadratic formula

Exam Questions – Solved by the quadratic formula

Introduction to the Quadratic Formula

  • The quadratic formula is an essential tool for solving quadratic equations. This formula is derived from the process of completing the square, and it can solve any quadratic equation.
  • It is written as: x = [-b ± sqrt(b² - 4ac)] / (2a), where a, b, and c are coefficients in the quadratic equation (ax² + bx + c = 0), and sqrt indicates the square root.

Using the Quadratic Formula

  • When given a quadratic equation, first identify and write down the values of a, b, and c.
  • Plug these values into the quadratic formula.
  • Simplify the expression inside the square root first, which is called the discriminant.
  • If the discriminant is positive, the equation will have two solutions. If it is zero, one solution. If it’s negative, the equation has no real solutions.
  • Finally, solve for x by performing the operations as indicated in the formula.

Solving Real-world Problems

  • Problems in real-life contexts can often be represented by quadratic equations.
  • For these problems, you will need to develop a quadratic equation from the problem statement and solve it using the quadratic formula.
  • Often, the solutions will represent certain quantities (like time, distance, or amount) that answer a question from the problem.

Solving the Quadratic Equation x^2 - 5x + 6 = 0 as an Example

  • Identify a, b, and c from the equation: a = 1, b = -5, c = 6.
  • Plug these values into the quadratic formula: x = [5 ± sqrt((-5)² - 4 * 1 * 6)] / (2 * 1).
  • Compute the discriminant (-5)² - 416 = 25 - 24 = 1.
  • Thus, the solutions are x = [5 + sqrt(1)] / 2 = 3 and x = [5 - sqrt(1)] / 2 = 2.

Key Points

  • The quadratic formula is essential for solving quadratic equations.
  • Make sure to correctly identify the coefficients a, b, and c from the equation.
  • Compute the discriminant to determine the number of solutions.
  • Real-world problems can often be modeled and solved using quadratic equations and the quadratic formula.