Quadratic Formula: Solving Quadratic Equations

To solve the quadratic equation ax^2 + bx + c = 0 using the quadratic formula, we can follow the steps below:

1. Identify the coefficients:
- a is the coefficient of x^2 term
- b is the coefficient of x term
- c is the constant term

2. Substitute the coefficients into the quadratic formula:
- The quadratic formula is given by x = (-b ± √(b^2 - 4ac)) / (2a)

3. Simplify the formula:
- Calculate the discriminant, which is the expression inside the square root: D = b^2 - 4ac

4. Calculate the square root of the discriminant:
- √D = √(b^2 - 4ac)

5. Calculate the solutions for x:
- Plug the values of √D, b, and a into the quadratic formula and solve for x.

6. Determine the nature of the solutions:
- If the discriminant D is positive, there are two distinct real solutions.
- If the discriminant D is zero, there is one real solution.
- If the discriminant D is negative, there are no real solutions, but there are two complex solutions.

Example:
Let's solve the quadratic equation 2x^2 + 5x - 3 = 0 using the quadratic formula.

1. Coefficients:
- a = 2
- b = 5
- c = -3

2. Quadratic Formula:
- x = (-b ± √(b^2 - 4ac)) / (2a)

3. Simplify the formula:
- Discriminant: D = b^2 - 4ac
D = (5^2) - (4 * 2 * -3)
D = 25 + 24
D = 49

4. Calculate the square root of the discriminant:
- √D = √49
- √D = 7

5. Calculate the solutions for x:
- x = (-5 ± 7) / (2 * 2)
For the positive root: x = (-5 + 7) / 4
For the negative root: x = (-5 - 7) / 4
Simplifying, we get:
- Positive root: x = 2 / 4
- Negative root: x = -12 / 4

6. Determine the nature of the solutions:
- Since the discriminant D is positive, we have two distinct real solutions.
- Positive root: x = 1/2
- Negative root: x = -3

Summary:
Using the quadratic formula, we can solve quadratic equations by substituting the coefficients into the formula and simplifying the equation step by step. The solution will depend on the value of the discriminant, which determines the nature of the solutions (real, complex, or none).