Use The Method Of Completing The Square To Solve The Quadratic Equation:${ 4x^2 - 9x + 5 = 0 }$
Introduction
Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. One of the most effective methods for solving quadratic equations is the method of completing the square. In this article, we will explore the method of completing the square and provide a step-by-step guide on how to use it to solve quadratic equations.
What is the Method of Completing the Square?
The method of 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 create a perfect square trinomial, which can be factored into the square of a binomial. This method is particularly useful when the quadratic equation does not factor easily.
Step 1: Write the Quadratic Equation in Standard Form
To begin, we need to write the quadratic equation in standard form, which is ax^2 + bx + c = 0. In this case, the equation is 4x^2 - 9x + 5 = 0.
Step 2: Move the Constant Term to the Right-Hand Side
Next, we need to move the constant term to the right-hand side of the equation. This will give us 4x^2 - 9x = -5.
Step 3: Divide Both Sides by the Coefficient of x^2
Now, we need to divide both sides of the equation by the coefficient of x^2, which is 4. This will give us x^2 - (9/4)x = -5/4.
Step 4: Add and Subtract the Square of Half the Coefficient of x
To complete the square, we need to add and subtract the square of half the coefficient of x. In this case, half the coefficient of x is -9/8, and its square is (-9/8)^2 = 81/64. We will add 81/64 to both sides of the equation.
Step 5: Factor the Perfect Square Trinomial
After adding and subtracting 81/64, we will have a perfect square trinomial on the left-hand side of the equation. We can factor this trinomial into the square of a binomial.
Step 6: Simplify the Equation
Finally, we need to simplify the equation by combining like terms.
Example: Solving the Quadratic Equation 4x^2 - 9x + 5 = 0
Let's apply the method of completing the square to the quadratic equation 4x^2 - 9x + 5 = 0.
Step 1: Write the Quadratic Equation in Standard Form
The equation is already in standard form: 4x^2 - 9x + 5 = 0.
Step 2: Move the Constant Term to the Right-Hand Side
Moving the constant term to the right-hand side gives us 4x^2 - 9x = -5.
Step 3: Divide Both Sides by the Coefficient of x^2
Dividing both sides by 4 gives us x^2 - (9/4)x = -5/4.
Step 4: Add and Subtract the Square of Half the Coefficient of x
Adding and subtracting 81/64 gives us x^2 - (9/4)x + 81/64 = -5/4 + 81/64.
Step 5: Factor the Perfect Square Trinomial
Factoring the perfect square trinomial gives us (x - 9/8)^2 = -5/4 + 81/64.
Step 6: Simplify the Equation
Simplifying the equation gives us (x - 9/8)^2 = 121/64.
Conclusion
The method of completing the square is a powerful technique for solving quadratic equations. By following the steps outlined in this article, you can use this method to solve quadratic equations of the form ax^2 + bx + c = 0. Remember to write the equation in standard form, move the constant term to the right-hand side, divide both sides by the coefficient of x^2, add and subtract the square of half the coefficient of x, factor the perfect square trinomial, and simplify the equation.
Common Mistakes to Avoid
When using the method of completing the square, there are several common mistakes to avoid. These include:
- Not writing the equation in standard form
- Not moving the constant term to the right-hand side
- Not dividing both sides by the coefficient of x^2
- Not adding and subtracting the square of half the coefficient of x
- Not factoring the perfect square trinomial
- Not simplifying the equation
Real-World Applications
The method of completing the square has several real-world applications. These include:
- Solving quadratic equations in physics and engineering
- Finding the maximum or minimum of a quadratic function
- Modeling population growth and decline
- Solving optimization problems
Conclusion
Q: What is the method of completing the square?
A: The method of 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 create a perfect square trinomial, which can be factored into the square of a binomial.
Q: When should I use the method of completing the square?
A: You should use the method of completing the square when the quadratic equation does not factor easily. This method is particularly useful when the quadratic equation has a complex or irrational solution.
Q: How do I know if the method of completing the square will work for my equation?
A: The method of completing the square will work for any quadratic equation of the form ax^2 + bx + c = 0, where a, b, and c are real numbers and a ≠0.
Q: What are the steps involved in the method of completing the square?
A: The steps involved in the method of completing the square are:
- Write the quadratic equation in standard form.
- Move the constant term to the right-hand side.
- Divide both sides by the coefficient of x^2.
- Add and subtract the square of half the coefficient of x.
- Factor the perfect square trinomial.
- Simplify the equation.
Q: What is the square of half the coefficient of x?
A: The square of half the coefficient of x is (b/2a)^2.
Q: Why do I need to add and subtract the square of half the coefficient of x?
A: You need to add and subtract the square of half the coefficient of x to create a perfect square trinomial.
Q: Can I use the method of completing the square to solve quadratic equations with complex coefficients?
A: Yes, you can use the method of completing the square to solve quadratic equations with complex coefficients.
Q: How do I know if the solution to the quadratic equation is real or complex?
A: You can determine if the solution to the quadratic equation is real or complex by checking the discriminant (b^2 - 4ac). If the discriminant is positive, the solution is real. If the discriminant is negative, the solution is complex.
Q: Can I use the method of completing the square to solve quadratic equations with irrational coefficients?
A: Yes, you can use the method of completing the square to solve quadratic equations with irrational coefficients.
Q: How do I know if the solution to the quadratic equation is rational or irrational?
A: You can determine if the solution to the quadratic equation is rational or irrational by checking the discriminant (b^2 - 4ac). If the discriminant is a perfect square, the solution is rational. If the discriminant is not a perfect square, the solution is irrational.
Q: Can I use the method of completing the square to solve quadratic equations with multiple solutions?
A: Yes, you can use the method of completing the square to solve quadratic equations with multiple solutions.
Q: How do I know if the quadratic equation has multiple solutions?
A: You can determine if the quadratic equation has multiple solutions by checking the discriminant (b^2 - 4ac). If the discriminant is zero, the equation has a repeated root. If the discriminant is positive, the equation has two distinct roots.
Conclusion
In conclusion, the method of completing the square is a powerful technique for solving quadratic equations. By following the steps outlined in this article, you can use this method to solve quadratic equations of the form ax^2 + bx + c = 0. Remember to write the equation in standard form, move the constant term to the right-hand side, divide both sides by the coefficient of x^2, add and subtract the square of half the coefficient of x, factor the perfect square trinomial, and simplify the equation.