Complete The Square Practice Worksheet
Completing the square is a technique used to solve quadratic equations of the form ax^2 + bx + c = 0. It involves manipulating the equation to express it in the form (x + d)^2 = e, where d and e are constants. This method is particularly useful when the quadratic expression cannot be factored easily.
Step 1: Identify the Quadratic Equation
To complete the square, we need to start with a quadratic equation in the form ax^2 + bx + c = 0.
- Example Equation: x^2 + 6x + 8 = 0
Step 2: Move the Constant Term to the Right Side
Move the constant term © to the right side of the equation to isolate the terms involving x.
- Resulting Equation: x^2 + 6x = -8
Step 3: Find the Value to Complete the Square
To complete the square, we need to find a value that, when squared, gives us the coefficient of the x term divided by 2. This value is then squared and added to both sides of the equation.
- Coefficient of x: 6
- Value to Complete the Square: (6 / 2)^2 = 3^2 = 9
Step 4: Add the Value to Both Sides
Add the value found in Step 3 to both sides of the equation to maintain equality.
- Resulting Equation: x^2 + 6x + 9 = -8 + 9
Step 5: Simplify the Right Side
Simplify the right side of the equation by performing the addition.
- Resulting Equation: x^2 + 6x + 9 = 1
Step 6: Factor the Left Side
Express the left side of the equation as a perfect square.
- Factored Form: (x + 3)^2 = 1
Step 7: Solve for x
Finally, solve for x by taking the square root of both sides of the equation.
- Solutions: x + 3 = ±1
- x Values: x = -3 ± 1
| Solutions | x Values |
|---|---|
| x + 3 = 1 | x = -2 |
| x + 3 = -1 | x = -4 |
📝 Note: Complete the square is a useful technique for solving quadratic equations that cannot be easily factored.
Practice Exercises:
- Exercise 1: Complete the square for the equation x^2 + 4x + 3 = 0.
- Exercise 2: Solve the quadratic equation x^2 - 2x - 5 = 0 using the complete the square method.
Solutions to Practice Exercises:
- Exercise 1 Solution: (x + 2)^2 - 1 = 0
- Exercise 2 Solution: (x - 1)^2 - 6 = 0
Complete the square is a powerful technique for solving quadratic equations. By following these steps and practicing with exercises, you can become proficient in using this method to solve a wide range of quadratic equations.
In essence, the process of completing the square involves manipulating a quadratic equation to express it in a perfect square form, making it easier to solve for the variable.
What is the purpose of completing the square?
+The purpose of completing the square is to express a quadratic equation in a perfect square form, making it easier to solve for the variable.
How do you find the value to complete the square?
+To find the value to complete the square, take the coefficient of the x term, divide it by 2, and then square the result.
What is the final step in solving a quadratic equation using the complete the square method?
+The final step is to take the square root of both sides of the equation and solve for x.
Related Terms:
- Remainder Theorem worksheet pdf
- Complex number Worksheet pdf
