QUadratic Formula Method

The quadratic formula is a general formula used to solve any quadratic equation of the form ax² + bx + c = 0. It provides a direct method for finding the solutions (roots) of a quadratic equation, regardless of whether it can be factored easily.

The formula is:

x = (-b ± √(b² - 4ac)) / (2a)

where:

• a, b, and c are the coefficients of the quadratic equation.

• x represents the solutions to the equation.

Steps to Use the Quadratic Formula:

1. Identify a, b, and c: Determine the values of a, b, and c from the given quadratic equation.

2. Substitute into the formula: Substitute the values of a, b, and c into the quadratic formula.

3. Simplify: Simplify the expression under the square root (the discriminant).

4. Calculate the solutions: Calculate the two possible values of x using the ± sign in the formula.

Example 1:

Solve the quadratic equation 2x² - 5x + 3 = 0 using the quadratic formula.

Steps:

1. Identify a, b, and c: a = 2, b = -5, c = 3

2. Substitute into the formula: x = (-(-5) ± √((-5)² - 423)) / (2*2)

3. Simplify: x = (5 ± √(25 - 24)) / 4

4. Calculate the solutions: x = (5 ± 1) / 4

   • x = 6/4 = 3/2

   • x = 4/4 = 1

Therefore, the solutions to the equation 2x² - 5x + 3 = 0 are x = 3/2 and x = 1.

Example 2:

Solve the quadratic equation 3x² - 7x + 2 = 0 using the quadratic formula.

Steps:

1. Identify a, b, and c: a = 3, b = -7, c = 2

2. Substitute into the formula: x = (-(-7) ± √((-7)² - 432)) / (2*3)

3. Simplify: x = (7 ± √(49 - 24)) / 6

4. Calculate the solutions: x = (7 ± √25) / 6

   • x = (7 + 5) / 6

   • x = 12/6 = 2

   • x = (7 - 5) / 6

   • x = 2/6 = 1/3

Therefore, the solutions to the equation 3x² - 7x + 2 = 0 are x = 2 and x = 1/3.

The quadratic formula is a versatile tool for solving quadratic equations, especially when factoring is difficult or impossible. It provides a direct and systematic approach to finding the solutions.