Solving Quadratic Equations using Quadratic Formula

Introduction

This tutorial will guide you through the process of solving quadratic equations using the quadratic formula. The quadratic formula is a powerful tool that allows you to find the roots of any quadratic equation in the form of ax² + bx + c = 0. Understanding how to apply this formula is essential for tackling various mathematical problems, especially in algebra.

Step 1: Identify the Coefficients

Before using the quadratic formula, you need to identify the coefficients of your quadratic equation.

• Write your equation in standard form: ax² + bx + c = 0.

Identify

• a (the coefficient of x²)
• b (the coefficient of x)
• c (the constant term)

Practical Tip:

Ensure the equation is equal to zero. If it isn't, rearrange it accordingly.

Step 2: Recall the Quadratic Formula

The quadratic formula is used to find the values of x that satisfy the equation. It is given by:

x = (-b ± √(b² - 4ac)) / (2a)

• Here, "±" indicates that there will generally be two solutions: one for addition and one for subtraction.

Step 3: Calculate the Discriminant

The discriminant (D) is the part of the formula under the square root, given by:

D = b² - 4ac

Determine the value of the discriminant

• If D > 0, there are two distinct real roots.
• If D = 0, there is one real root (a repeated root).
• If D < 0, there are no real roots (the roots are complex).

Common Pitfall:

Always calculate the discriminant before proceeding to find the roots, as it tells you the nature of the roots.

Step 4: Substitute Values into the Formula

Insert the values of a, b, and c into the quadratic formula.

• Calculate the square root of the discriminant.
• Substitute the values into the formula:

x = (-b ± √D) / (2a)

Step 5: Simplify the Results

After substituting the values, simplify the expression to find the values of x.

• Perform the addition and subtraction to find both potential solutions.
• Ensure to simplify any fractions or square roots as necessary.

Practical Tip:

Double-check your calculations to avoid simple arithmetic errors.

Conclusion

You have now learned how to solve quadratic equations using the quadratic formula. Remember to identify the coefficients, calculate the discriminant, and then substitute and simplify your results. Practicing this method will enhance your proficiency in solving quadratic equations, which is essential for advancing in algebra and other areas of mathematics. For further practice, try solving different quadratic equations using the steps outlined in this guide.