Form The Perfect Square Trinomial In The Process Of Completing The Square. What Is The Value Of $c$? + 3 X + C = 7 4 + C = □ \begin{array}{l} +3x+c=\frac{7}{4}+c \\ =\square \end{array} + 3 X + C = 4 7 + C = □
Introduction
Completing the square is a powerful algebraic technique used to solve quadratic equations and express them in a perfect square trinomial form. This method involves manipulating the quadratic equation to create a perfect square trinomial on one side of the equation, which can then be factored into the square of a binomial. In this article, we will explore the process of completing the square and find the value of c in the given equation.
What is Completing the Square?
Completing the square is a method of solving quadratic equations by manipulating the equation to create a perfect square trinomial on one side. This involves adding and subtracting a constant term to create a perfect square trinomial, which can then be factored into the square of a binomial. The general form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants.
The Process of Completing the Square
To complete the square, we follow these steps:
- Write the quadratic equation in the form ax^2 + bx + c = 0.
- Move the constant term to the right-hand side of the equation.
- Add and subtract (b/2)^2 to the left-hand side of the equation.
- Factor the left-hand side of the equation into the square of a binomial.
- Simplify the right-hand side of the equation.
Example: Completing the Square
Let's consider the quadratic equation x^2 + 3x + c = 7/4 + c. Our goal is to complete the square and find the value of c.
Step 1: Write the quadratic equation in the form ax^2 + bx + c = 0
x^2 + 3x + c = 7/4 + c
Step 2: Move the constant term to the right-hand side of the equation
x^2 + 3x = 7/4
Step 3: Add and subtract (b/2)^2 to the left-hand side of the equation
x^2 + 3x + (3/2)^2 = (3/2)^2 + 7/4
Step 4: Factor the left-hand side of the equation into the square of a binomial
(x + 3/2)^2 = (3/2)^2 + 7/4
Step 5: Simplify the right-hand side of the equation
(x + 3/2)^2 = 9/4 + 7/4
(x + 3/2)^2 = 16/4
(x + 3/2)^2 = 4
Finding the Value of c
Now that we have completed the square, we can find the value of c by equating the two expressions.
(x + 3/2)^2 = 4
x^2 + 3x + (3/2)^2 = 4
x^2 + 3x + 9/4 = 4
x^2 + 3x = 4 - 9/4
x^2 + 3x = 7/4 - 9/4
x^2 + 3x = -2/4
x^2 + 3x = -1/2
x^2 + 3x c = -1/2
c = -1/2 - (x^2 + 3x)
c = -1/2 - x^2 - 3x
c = -1/2 - (x^2 + 3x + (3/2)^2) + (3/2)^2
c = -1/2 - (x + 3/2)^2 + 9/4
c = -1/2 - 4 + 9/4
c = -1/2 - 16/4 + 9/4
c = -1/2 - 7/4
c = -2/4 - 7/4
c = -9/4
Conclusion
In this article, we have explored the process of completing the square and found the value of c in the given equation. Completing the square is a powerful algebraic technique used to solve quadratic equations and express them in a perfect square trinomial form. By following the steps outlined in this article, we can complete the square and find the value of c in any quadratic equation.
Frequently Asked Questions
- What is completing the square? Completing the square is a method of solving quadratic equations by manipulating the equation to create a perfect square trinomial on one side.
- How do I complete the square? To complete the square, follow these steps: write the quadratic equation in the form ax^2 + bx + c = 0, move the constant term to the right-hand side of the equation, add and subtract (b/2)^2 to the left-hand side of the equation, factor the left-hand side of the equation into the square of a binomial, and simplify the right-hand side of the equation.
- What is the value of c in the given equation? The value of c in the given equation is -9/4.
References
- [1] "Completing the Square" by Math Open Reference
- [2] "Quadratic Equations" by Khan Academy
- [3] "Completing the Square" by Purplemath
Introduction
Completing the square is a powerful algebraic technique used to solve quadratic equations and express them in a perfect square trinomial form. In our previous article, we explored the process of completing the square and found the value of c in the given equation. In this article, we will provide a Q&A guide to help you better understand the concept of completing the square.
Q&A Guide
Q: What is completing the square?
A: Completing the square is a method of solving quadratic equations by manipulating the equation to create a perfect square trinomial on one side.
Q: How do I complete the square?
A: To complete the square, follow these steps:
- Write the quadratic equation in the form ax^2 + bx + c = 0.
- Move the constant term to the right-hand side of the equation.
- Add and subtract (b/2)^2 to the left-hand side of the equation.
- Factor the left-hand side of the equation into the square of a binomial.
- Simplify the right-hand side of the equation.
Q: What is the purpose of completing the square?
A: The purpose of completing the square is to express a quadratic equation in a perfect square trinomial form, which can be factored into the square of a binomial.
Q: How do I determine the value of c in a quadratic equation?
A: To determine the value of c in a quadratic equation, you can complete the square and then equate the two expressions.
Q: What is the value of c in the given equation?
A: The value of c in the given equation is -9/4.
Q: Can I use completing the square to solve quadratic equations with complex coefficients?
A: Yes, you can use completing the square to solve quadratic equations with complex coefficients.
Q: How do I apply completing the square to quadratic equations with rational coefficients?
A: To apply completing the square to quadratic equations with rational coefficients, follow the same steps as before, but be sure to simplify the expression carefully.
Q: Can I use completing the square to solve quadratic equations with irrational coefficients?
A: Yes, you can use completing the square to solve quadratic equations with irrational coefficients.
Q: How do I determine the number of solutions to a quadratic equation?
A: To determine the number of solutions to a quadratic equation, you can use the discriminant, which is the expression under the square root in the quadratic formula.
Q: What is the discriminant?
A: The discriminant is the expression under the square root in the quadratic formula, which is b^2 - 4ac.
Q: How do I use the discriminant to determine the number of solutions to a quadratic equation?
A: To use the discriminant to determine the number of solutions to a quadratic equation, follow these steps:
- Calculate the discriminant.
- If the discriminant is positive, the equation has two distinct real solutions.
- If the discriminant is zero, the equation has one real solution.
- If the discriminant is negative, the equation has no real solutions.
Conclusion
In this article, we have provided a Q&A guide to help you better understand the concept of completing the square. Completing the square is powerful algebraic technique used to solve quadratic equations and express them in a perfect square trinomial form. By following the steps outlined in this article, you can complete the square and determine the value of c in any quadratic equation.
Frequently Asked Questions
- What is completing the square? Completing the square is a method of solving quadratic equations by manipulating the equation to create a perfect square trinomial on one side.
- How do I complete the square? To complete the square, follow these steps: write the quadratic equation in the form ax^2 + bx + c = 0, move the constant term to the right-hand side of the equation, add and subtract (b/2)^2 to the left-hand side of the equation, factor the left-hand side of the equation into the square of a binomial, and simplify the right-hand side of the equation.
- What is the value of c in the given equation? The value of c in the given equation is -9/4.
References
- [1] "Completing the Square" by Math Open Reference
- [2] "Quadratic Equations" by Khan Academy
- [3] "Completing the Square" by Purplemath