To find the numbers, or more accurately, the roots or solutions of a quadratic equation, you can use the quadratic formula.
Understanding Quadratic Equations
A quadratic equation is a polynomial equation of the second degree. The standard form of a quadratic equation is:
ax² + bx + c = 0
Where:
- 'a', 'b', and 'c' are coefficients, with 'a' not equal to zero.
- 'x' is the variable.
The Quadratic Formula: Your Key to Finding Solutions
The quadratic formula is a tool that helps us find the values of 'x' that satisfy the quadratic equation. It's derived by completing the square in the standard quadratic form and is expressed as:
x = (-b ± √(b² - 4ac)) / 2a
Here's how to apply this formula:
- Identify the Coefficients: First, identify 'a', 'b', and 'c' from your quadratic equation. Make sure the equation is in the standard form ax² + bx + c = 0.
- Plug into the Formula: Substitute the values of 'a', 'b', and 'c' into the quadratic formula.
- Solve for x: Simplify the expression and you'll get two possible solutions for 'x', due to the ± symbol (plus and minus). These are the roots of your quadratic equation.
Practical Example
Let's consider the equation 2x² + 5x - 3 = 0. In this equation:
- a = 2
- b = 5
- c = -3
Now we substitute these values into the formula:
x = (-5 ± √(5² - 4 2 -3)) / (2 * 2)
x = (-5 ± √(25 + 24)) / 4
x = (-5 ± √49) / 4
x = (-5 ± 7) / 4
This gives us two possible solutions:
- x₁ = (-5 + 7) / 4 = 2/4 = 1/2
- x₂ = (-5 - 7) / 4 = -12/4 = -3
Thus, the solutions, or numbers of this quadratic equation are x=1/2 and x=-3.
Steps to Using the Quadratic Formula:
- Write the equation in standard form: ax² + bx + c = 0
- Identify a, b, and c. Make sure 'a', 'b', and 'c' are correctly identified.
- Plug a, b, and c into the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a
- Solve for x. Calculate the two solutions by first using the "+" sign, then the "-" sign.
Key Takeaways
- The quadratic formula provides a direct way to find the solutions of any quadratic equation.
- It is crucial to have the quadratic equation in standard form (ax² + bx + c = 0) before using the formula.
- Remember to handle the ± sign to obtain the two solutions.
- The values of a,b, and c are critical to the final solution.
