Rewrite The Function By Completing The Square.${ \begin{array}{l} g(x) = X^2 - X - 6 \ g(x) = \square(x + \square)^2 + \square \end{array} }$

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Introduction

Completing the square is a powerful technique used in algebra to rewrite quadratic expressions in a more convenient form. This method involves manipulating the quadratic expression to express it as a perfect square trinomial, which can be factored into the square of a binomial. In this article, we will focus on rewriting the function g(x) = x^2 - x - 6 by completing the square.

What is Completing the Square?

Completing the square is a process of rewriting a quadratic expression in the form (x + a)^2 + b, where a and b are constants. This form is useful because it allows us to easily identify the vertex of the parabola represented by the quadratic expression. The vertex form of a quadratic function is given by f(x) = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.

Step 1: Identify the Coefficients

To complete the square, we need to identify the coefficients of the quadratic expression g(x) = x^2 - x - 6. The coefficient of the x^2 term is 1, the coefficient of the x term is -1, and the constant term is -6.

Step 2: Move the Constant Term

The next step is to move the constant term to the right-hand side of the equation. This gives us g(x) = x^2 - x = 6.

Step 3: Find the Value to Add

To complete the square, we need to find the value to add to the left-hand side of the equation. This value is given by the formula a^2/4, where a is the coefficient of the x term. In this case, a = -1, so the value to add is (-1)^2/4 = 1/4.

Step 4: Add and Subtract the Value

We add and subtract the value to the left-hand side of the equation: g(x) = x^2 - x + 1/4 - 1/4 = 6.

Step 5: Factor the Perfect Square

The left-hand side of the equation is now a perfect square trinomial: g(x) = (x - 1/2)^2 - 1/4 = 6. We can factor the perfect square trinomial as follows: g(x) = (x - 1/2)^2 - 1/4 = (x - 1/2)^2 - 1/4 = (x - 1/2)^2 - 1/4 = (x - 1/2)^2 - 1/4 = (x - 1/2)^2 - 1/4.

Step 6: Simplify the Expression

We can simplify the expression by combining the constant terms: g(x) = (x - 1/2)^2 - 1/4 = (x - 1/2)^2 - 1/4 = (x - 1/2)^2 - 1/4 = (x - 1/2)^2 - 1/4.

The Final Answer

The final answer is g(x) = (x - 1/2)^2 - 1/4 = 6. This is the rewritten function in the form (x + a)^2 + b, where a = -1/2 and b = -1/4.

Conclusion

Completing the square is a powerful technique used in algebra to rewrite quadratic expressions in a more convenient form. By following the steps outlined in this article, we can rewrite the function g(x) = x^2 - x - 6 in the form (x + a)^2 + b. This form is useful because it allows us to easily identify the vertex of the parabola represented by the quadratic expression.

Example Problems

Here are some example problems that illustrate the concept of completing the square:

Example 1

Rewrite the function f(x) = x^2 + 4x + 4 by completing the square.

Solution

To complete the square, we need to identify the coefficients of the quadratic expression. The coefficient of the x^2 term is 1, the coefficient of the x term is 4, and the constant term is 4. We can move the constant term to the right-hand side of the equation: f(x) = x^2 + 4x = -4. We can find the value to add by using the formula a^2/4, where a is the coefficient of the x term. In this case, a = 4, so the value to add is 4^2/4 = 4. We can add and subtract the value to the left-hand side of the equation: f(x) = x^2 + 4x + 4 - 4 = -4. We can factor the perfect square trinomial as follows: f(x) = (x + 2)^2 - 4 = -4. We can simplify the expression by combining the constant terms: f(x) = (x + 2)^2 - 4 = -4.

Example 2

Rewrite the function g(x) = x^2 - 2x - 3 by completing the square.

Solution

To complete the square, we need to identify the coefficients of the quadratic expression. The coefficient of the x^2 term is 1, the coefficient of the x term is -2, and the constant term is -3. We can move the constant term to the right-hand side of the equation: g(x) = x^2 - 2x = 3. We can find the value to add by using the formula a^2/4, where a is the coefficient of the x term. In this case, a = -2, so the value to add is (-2)^2/4 = 1. We can add and subtract the value to the left-hand side of the equation: g(x) = x^2 - 2x + 1 - 1 = 3. We can factor the perfect square trinomial as follows: g(x) = (x - 1)^2 - 1 = 3. We can simplify the expression by combining the constant terms: g(x) = (x - 1)^2 - 1 = 3.

Final Thoughts

Q: What is completing the square?

A: Completing the square is a technique used in algebra to rewrite quadratic expressions in a more convenient form. It involves manipulating the quadratic expression to express it as a perfect square trinomial, which can be factored into the square of a binomial.

Q: Why is completing the square useful?

A: Completing the square is useful because it allows us to easily identify the vertex of the parabola represented by the quadratic expression. It also helps us to solve quadratic equations and find the roots of the equation.

Q: How do I complete the square?

A: To complete the square, you need to follow these steps:

  1. Identify the coefficients of the quadratic expression.
  2. Move the constant term to the right-hand side of the equation.
  3. Find the value to add by using the formula a^2/4, where a is the coefficient of the x term.
  4. Add and subtract the value to the left-hand side of the equation.
  5. Factor the perfect square trinomial.

Q: What is the value to add?

A: The value to add is given by the formula a^2/4, where a is the coefficient of the x term.

Q: How do I find the value to add?

A: To find the value to add, you need to square the coefficient of the x term and divide it by 4.

Q: What is the difference between completing the square and factoring?

A: Completing the square and factoring are two different techniques used to rewrite quadratic expressions. Factoring involves expressing the quadratic expression as a product of two binomials, while completing the square involves expressing it as a perfect square trinomial.

Q: Can I use completing the square to solve quadratic equations?

A: Yes, you can use completing the square to solve quadratic equations. By rewriting the quadratic expression in the form (x + a)^2 + b, you can easily identify the roots of the equation.

Q: How do I use completing the square to solve quadratic equations?

A: To use completing the square to solve quadratic equations, you need to follow these steps:

  1. Rewrite the quadratic expression in the form (x + a)^2 + b.
  2. Set the expression equal to zero.
  3. Solve for x.

Q: What are some common mistakes to avoid when completing the square?

A: Some common mistakes to avoid when completing the square include:

  • Not moving the constant term to the right-hand side of the equation.
  • Not finding the value to add correctly.
  • Not adding and subtracting the value correctly.
  • Not factoring the perfect square trinomial correctly.

Q: How do I know if I have completed the square correctly?

A: To know if you have completed the square correctly, you need to check that the expression is in the form (x + a)^2 + b. You can also check that the expression can be factored into the square of a binomial.

Q: Can I use completing the square to solve quadratic equations with complex roots?

A: Yes, you can use completing the square to solve quadratic equations with complex roots. By rewriting the quadratic expression in the form (x + a)^2 + b, you can easily identify the roots of the equation, including complex roots.

Q: How do I use completing the square to solve quadratic equations with complex roots?

A: To use completing the square to solve quadratic equations with complex roots, you need to follow the same steps as before:

  1. Rewrite the quadratic expression in the form (x + a)^2 + b.
  2. Set the expression equal to zero.
  3. Solve for x.

Conclusion

Completing the square is a powerful technique used in algebra to rewrite quadratic expressions in a more convenient form. By following the steps outlined in this article, you can complete the square and solve quadratic equations. Remember to avoid common mistakes and check your work to ensure that you have completed the square correctly.