How To Solve Quadratic Equations Using The Quadratic Formula

Introduction

This tutorial will guide you through solving quadratic equations using the quadratic formula. The quadratic formula is a powerful tool that helps find the roots of any quadratic equation in the form (ax^2 + bx + c = 0). Understanding this method is essential for solving various mathematical problems, particularly in algebra and calculus.

Step 1: Identify Coefficients

Before applying the quadratic formula, you need to identify the coefficients (a), (b), and (c) from the equation in standard form.

• For the equation (2x^2 + 3x - 2 = 0):
  • (a = 2) (coefficient of (x^2))
  • (b = 3) (coefficient of (x))
  • (c = -2) (constant term)

Step 2: Write the Quadratic Formula

The quadratic formula is given by:

[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ]

This formula allows you to calculate the values of (x) that satisfy the equation.

Step 3: Calculate the Discriminant

The discriminant is the part of the formula under the square root, (b^2 - 4ac). It helps determine the nature of the roots.

• For our example:
  • Calculate (b^2 = 3^2 = 9)
  • Calculate (4ac = 4 \cdot 2 \cdot (-2) = -16)
  • Therefore, the discriminant is (9 - (-16) = 25)

Step 4: Evaluate the Square Root

Now, find the square root of the discriminant.

• The square root of (25) is (5).

Step 5: Substitute into the Quadratic Formula

Substitute the values of (b) and the square root of the discriminant back into the quadratic formula.

• For our equation:

[ x = \frac{-3 \pm 5}{4} ]

Step 6: Solve for the Roots

Now, break it down into two separate calculations due to the (\pm) symbol.

1. First Root:

   • (x_1 = \frac{-3 + 5}{4} = \frac{2}{4} = \frac{1}{2})

2. Second Root:

   • (x_2 = \frac{-3 - 5}{4} = \frac{-8}{4} = -2)

Step 7: Verify the Solutions

To confirm that your solutions are correct, substitute them back into the original equation.

• For (x = -2):
  • (2(-2)^2 + 3(-2) - 2 = 0)
  • (8 - 6 - 2 = 0)

The solution is verified as correct.

Conclusion

You have learned how to solve quadratic equations using the quadratic formula by identifying coefficients, calculating the discriminant, and substituting values into the formula. To further enhance your understanding, consider practicing with different quadratic equations or exploring methods such as factoring or completing the square.