Rewrite The Function By Completing The Square.$\[ F(x) = X^2 - 10x - 96 \\]$\[ F(x) = (x + \square)^2 + \square \\]

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Introduction

Completing the square is a powerful technique used in algebra to rewrite quadratic functions in a more convenient form. This method involves expressing a quadratic function as a perfect square trinomial, which can be useful in solving equations, graphing functions, and simplifying expressions. In this article, we will focus on rewriting the given function by completing the square.

The Given Function

The given function is:

f(x)=x2−10x−96{ f(x) = x^2 - 10x - 96 }

Our goal is to rewrite this function in the form:

f(x)=(x+â–¡)2+â–¡{ f(x) = (x + \square)^2 + \square }

Step 1: Move the Constant Term

To complete the square, we need to move the constant term to the right-hand side of the equation. This will give us:

x2−10x=96{ x^2 - 10x = 96 }

Step 2: Find the Value to Add

To make the left-hand side a perfect square trinomial, we need to add a value to both sides of the equation. This value is half of the coefficient of the x-term squared. In this case, the coefficient of the x-term is -10, so we need to add:

(−102)2=25{ \left( \frac{-10}{2} \right)^2 = 25 }

to both sides of the equation.

Step 3: Add the Value to Both Sides

Adding 25 to both sides of the equation gives us:

x2−10x+25=96+25{ x^2 - 10x + 25 = 96 + 25 }

Simplifying the right-hand side, we get:

x2−10x+25=121{ x^2 - 10x + 25 = 121 }

Step 4: Factor the Perfect Square Trinomial

The left-hand side of the equation is now a perfect square trinomial, which can be factored as:

(x−5)2=121{ (x - 5)^2 = 121 }

Step 5: Rewrite the Function

Finally, we can rewrite the original function in the desired form:

f(x)=(x−5)2−121{ f(x) = (x - 5)^2 - 121 }

However, we can simplify this further by adding 121 to both sides of the equation:

f(x)=(x−5)2{ f(x) = (x - 5)^2 }

Conclusion

In this article, we have successfully rewritten the given function by completing the square. We moved the constant term to the right-hand side, found the value to add, added the value to both sides, factored the perfect square trinomial, and finally rewrote the function in the desired form. This technique is a powerful tool in algebra and can be used to solve equations, graph functions, and simplify expressions.

Example 1: Solving an Equation

Suppose we want to solve the equation:

x2−10x−96=0{ x^2 - 10x - 96 = 0 }

Using the technique of completing the square, we can rewrite the equation as:

(x−5)2=121{ (x - 5)^2 = 121 }

Taking the square root of both sides, we get:

x−5=±121{ x - 5 = \pm \sqrt{121} }

Simplifying, we get:

x−5=±11{ x - 5 = \pm 11 }

Adding 5 to both sides, we get:

x=5±11{ x = 5 \pm 11 }

Simplifying, we get:

x=−6 or x=16{ x = -6 \text{ or } x = 16 }

Therefore, the solutions to the equation are x = -6 and x = 16.

Example 2: Graphing a Function

Suppose we want to graph the function:

f(x)=x2−10x−96{ f(x) = x^2 - 10x - 96 }

Using the technique of completing the square, we can rewrite the function as:

f(x)=(x−5)2−121{ f(x) = (x - 5)^2 - 121 }

This function is a parabola that opens upwards, with a vertex at (5, -121). The graph of the function is a U-shaped curve that is shifted 5 units to the right and 121 units downwards.

Conclusion

Introduction

Completing the square is a powerful technique used in algebra to rewrite quadratic functions in a more convenient form. In our previous article, we discussed the steps involved in completing the square and provided examples of solving equations and graphing functions using this technique. In this article, we will answer some frequently asked questions about completing the square.

Q: What is completing the square?

A: Completing the square is a technique used to rewrite a quadratic function in the form (x + a)^2 + b, where a and b are constants. This form is useful in solving equations, graphing functions, and simplifying expressions.

Q: Why is completing the square useful?

A: Completing the square is useful because it allows us to rewrite a quadratic function in a more convenient form. This form is useful in solving equations, graphing functions, and simplifying expressions. It also helps us to identify the vertex of a parabola and the direction in which it opens.

Q: How do I know when to use completing the square?

A: You should use completing the square when you are given a quadratic function in the form ax^2 + bx + c and you want to rewrite it in the form (x + a)^2 + b. This technique is particularly useful when you are solving equations or graphing functions.

Q: What are the steps involved in completing the square?

A: The steps involved in completing the square are:

  1. Move the constant term to the right-hand side of the equation.
  2. Find the value to add to both sides of the equation.
  3. Add the value to both sides of the equation.
  4. Factor the perfect square trinomial.
  5. Rewrite the function in the desired form.

Q: How do I find the value to add?

A: To find the value to add, you need to take half of the coefficient of the x-term and square it. This value is then added to both sides of the equation.

Q: What if I have a quadratic function with a negative coefficient of x^2?

A: If you have a quadratic function with a negative coefficient of x^2, you can rewrite it as a perfect square trinomial by factoring out the negative sign. For example, if you have the function x^2 - 10x - 96, you can rewrite it as (x - 5)^2 - 121.

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 equation in the form (x + a)^2 + b, you can easily identify the solutions to the equation.

Q: How do I graph a quadratic function using completing the square?

A: To graph a quadratic function using completing the square, you need to rewrite the function in the form (x + a)^2 + b. The graph of the function is a parabola that opens upwards or downwards, depending on the sign of the coefficient of x^2.

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 correct value to add.
  • Not adding the value to both sides of the equation.
  • Not factoring the perfect square trinomial correctly.

Conclusion

In conclusion, completing the square is a powerful technique used in algebra to rewrite quadratic functions in a more convenient form. By following the steps involved in completing the square and avoiding common mistakes, you can easily rewrite quadratic functions and solve equations, graph functions, and simplify expressions.