How is the quadratic formula applied?

Short Answer

The quadratic formula is applied by substituting the values of a, b, and c from the quadratic equation into the formula x = (−b ± √(b² − 4ac)) / 2a. This helps in finding the solutions of the equation step by step.

First, we write the equation in standard form, then identify the values, and finally calculate the result using the formula. It gives two possible answers for the variable.

Detailed Explanation

Applying quadratic formula

To apply the quadratic formula, we use it on a quadratic equation written in the standard form:

The formula used is:

This formula helps us find the value of x, which are the solutions (roots) of the equation.

Steps to apply the formula

Follow these simple steps to apply the quadratic formula:

  1. Write the equation in standard form
    Make sure the equation is written as ax² + bx + c = 0.
  2. Identify coefficients
    Find the values of a, b, and c from the equation.
  3. Substitute values
    Put the values into the formula.
  4. Calculate the discriminant
    Find the value of b² − 4ac.
  5. Find square root
    Take the square root of the discriminant.
  6. Solve for x
    Use both + and − signs to find two values of x.

Example for understanding

Let us solve the equation:

x² − 4x − 5 = 0

Step 1: Identify values
a = 1, b = −4, c = −5

Step 2: Substitute into formula
x = (−(−4) ± √((−4)² − 4×1×(−5))) / 2×1

Step 3: Simplify
x = (4 ± √(16 + 20)) / 2
x = (4 ± √36) / 2

Step 4: Solve
x = (4 + 6) / 2 = 5
x = (4 − 6) / 2 = −1

So, the solutions are x = 5 and x = −1.

Importance of applying formula

Applying the quadratic formula is very useful because:

  • It works for all types of quadratic equations
  • It gives exact answers
  • It is helpful when factorization is not possible
  • It reduces confusion in solving equations

It is a step-by-step method that ensures accuracy.

Common mistakes to avoid

While applying the quadratic formula, students should be careful about:

  • Writing the correct values of a, b, and c
  • Paying attention to signs (positive and negative)
  • Correct calculation of square root
  • Using both + and − signs

Avoiding these mistakes helps in getting the correct answer.

Conclusion

The quadratic formula is applied by carefully substituting the values of a, b, and c into the formula and solving step by step. It is a reliable and universal method for solving quadratic equations. Learning how to apply it correctly makes solving equations easier and more accurate.

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