Brian Is Solving The Equation { X^2 - X - 5$}$. What Value Must Be Added To Both Sides Of The Equation To Make The Left Side A Perfect-square Trinomial?A. 16 B. 4 C. 9 D. 6

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Introduction

In algebra, a perfect-square trinomial is a polynomial expression that can be factored into the square of a binomial. It is a crucial concept in solving quadratic equations. In this article, we will explore how to solve the equation {x^2 - x - 5$}$ by making the left side a perfect-square trinomial.

Understanding Perfect-Square Trinomials

A perfect-square trinomial is a quadratic expression that can be written in the form (a+b)2{(a + b)^2} or (a−b)2{(a - b)^2}. It has a specific pattern, where the first and last terms are perfect squares, and the middle term is twice the product of the square roots of the first and last terms.

The General Form of a Perfect-Square Trinomial

The general form of a perfect-square trinomial is a2+2ab+b2{a^2 + 2ab + b^2} or a2−2ab+b2{a^2 - 2ab + b^2}. To make the left side of the equation {x^2 - x - 5$}$ a perfect-square trinomial, we need to add a value to both sides of the equation.

Finding the Value to be Added

To find the value to be added, we need to complete the square. We start by taking half of the coefficient of the middle term, which is -1 in this case. Half of -1 is -0.5. We then square this value to get 0.25.

The Formula for Completing the Square

The formula for completing the square is a2+2ab+b2=(a+b)2{a^2 + 2ab + b^2 = (a + b)^2}. We can use this formula to rewrite the left side of the equation {x^2 - x - 5$}$ as a perfect-square trinomial.

Rewriting the Equation

To rewrite the equation {x^2 - x - 5$}$ as a perfect-square trinomial, we need to add 0.25 to both sides of the equation. This will make the left side a perfect-square trinomial.

The Final Answer

The value that must be added to both sides of the equation to make the left side a perfect-square trinomial is 0.25. However, this is not among the answer choices. We need to find the closest answer choice.

Comparing the Answer Choices

Q: What is a perfect-square trinomial?

A: A perfect-square trinomial is a polynomial expression that can be factored into the square of a binomial. It is a crucial concept in solving quadratic equations.

Q: How do I know if an expression is a perfect-square trinomial?

A: To determine if an expression is a perfect-square trinomial, you need to check if it can be written in the form (a+b)2{(a + b)^2} or (a−b)2{(a - b)^2}. If it can be written in this form, then it is a perfect-square trinomial.

Q: What is the general form of a perfect-square trinomial?

A: The general form of a perfect-square trinomial is a2+2ab+b2{a^2 + 2ab + b^2} or a2−2ab+b2{a^2 - 2ab + b^2}.

Q: How do I complete the square to make the left side of the equation a perfect-square trinomial?

A: To complete the square, you need to take half of the coefficient of the middle term and square it. Then, add this value to both sides of the equation.

Q: What is the formula for completing the square?

A: The formula for completing the square is a2+2ab+b2=(a+b)2{a^2 + 2ab + b^2 = (a + b)^2}.

Q: How do I find the value to be added to both sides of the equation to make the left side a perfect-square trinomial?

A: To find the value to be added, you need to take half of the coefficient of the middle term and square it.

Q: What is the value that must be added to both sides of the equation to make the left side a perfect-square trinomial?

A: The value that must be added to both sides of the equation to make the left side a perfect-square trinomial is 0.25.

Q: Why is 0.25 not among the answer choices?

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