Rewrite The Equation In The Form $(x-p)^2=q$.$x^2 + 5x + \frac{5}{4} = 0$

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Introduction

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Quadratic equations are a fundamental concept in mathematics, and rewriting them in a specific form can be a crucial step in solving them. In this article, we will focus on rewriting the equation $x^2 + 5x + \frac{5}{4} = 0$ in the form $(x-p)^2=q$. This form is known as the vertex form of a quadratic equation, where $p$ represents the x-coordinate of the vertex and $q$ represents the y-coordinate of the vertex.

Understanding the Vertex Form

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The vertex form of a quadratic equation is given by the equation $(x-p)^2=q$, where $p$ and $q$ are constants. This form is useful because it allows us to easily identify the vertex of the parabola represented by the quadratic equation. The vertex form can be obtained by completing the square, which involves rewriting the quadratic equation in a way that allows us to easily identify the vertex.

Completing the Square

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To rewrite the equation $x^2 + 5x + \frac{5}{4} = 0$ in the form $(x-p)^2=q$, we need to complete the square. This involves adding and subtracting a constant term to the equation, which allows us to rewrite it in the desired form.

Step 1: Move the Constant Term to the Right-Hand Side

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The first step in completing the square is to move the constant term to the right-hand side of the equation. This gives us:

$$x^2 + 5x = -\frac{5}{4}$$

Step 2: Add and Subtract a Constant Term

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Next, we need to add and subtract a constant term to the left-hand side of the equation. The constant term we add is the square of half the coefficient of the $x$ term. In this case, the coefficient of the $x$ term is 5, so we add and subtract $\left(\frac{5}{2}\right)^2 = \frac{25}{4}$:

$$x^2 + 5x + \frac{25}{4} - \frac{25}{4} = -\frac{5}{4}$$

Step 3: Rewrite the Left-Hand Side as a Perfect Square

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Now that we have added and subtracted the constant term, we can rewrite the left-hand side of the equation as a perfect square:

$$(x + \frac{5}{2})^2 - \frac{25}{4} = -\frac{5}{4}$$

Step 4: Simplify the Equation

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Finally, we can simplify the equation by adding $\frac{25}{4}$ to both sides:

$$(x + \frac{5}{2})^2 = \frac{25}{4}$$

Conclusion

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In this article, we have rewritten the equation $x^2 + 5x + \frac{5}{4} = 0$ in the form $(x-p)^2=q$. This form is known as the vertex form of a quadratic equation, where $p$ represents the x-coordinate of the vertex and $q$ represents the y-coordinate of the vertex. We have completed the square to obtain the vertex form, which involves adding and subtracting a constant term to the equation. This form is useful because it allows us to easily identify the vertex of the parabola represented by the quadratic equation.

Example Problems

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Problem 1

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Rewrite the equation $x^2 - 6x + 8 = 0$ in the form $(x-p)^2=q$.

Solution

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To rewrite the equation $x^2 - 6x + 8 = 0$ in the form $(x-p)^2=q$, we need to complete the square. This involves adding and subtracting a constant term to the equation, which allows us to rewrite it in the desired form.

Step 1: Move the Constant Term to the Right-Hand Side

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The first step in completing the square is to move the constant term to the right-hand side of the equation. This gives us:

$$x^2 - 6x = -8$$

Step 2: Add and Subtract a Constant Term

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Next, we need to add and subtract a constant term to the left-hand side of the equation. The constant term we add is the square of half the coefficient of the $x$ term. In this case, the coefficient of the $x$ term is -6, so we add and subtract $\left(\frac{-6}{2}\right)^2 = 9$:

$$x^2 - 6x + 9 - 9 = -8$$

Step 3: Rewrite the Left-Hand Side as a Perfect Square

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Now that we have added and subtracted the constant term, we can rewrite the left-hand side of the equation as a perfect square:

$$(x - 3)^2 - 9 = -8$$

Step 4: Simplify the Equation

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Finally, we can simplify the equation by adding 9 to both sides:

$$(x - 3)^2 = 1$$

Problem 2

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Rewrite the equation $x^2 + 2x - 3 = 0$ in the form $(x-p)^2=q$.

Solution

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To rewrite the equation $x^2 + 2x - 3 = 0$ in the form $(x-p)^2=q$, we need to complete the square. This involves adding and subtracting a constant term to the equation, which allows us to rewrite it in the desired form.

Step 1: Move the Constant Term to the Right-Hand Side

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The first step in completing the square is to move the constant term to the right-hand side of the equation. This gives us:

$$x^2 + 2x = 3$$

Step 2: Add and Subtract a Constant Term

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Next, we need to add and subtract a constant term to the left-hand side of the equation. The constant term we add is the square of half the coefficient of the $x$ term. In this case, the coefficient of the $x$ term is 2, so we add and subtract $\left(\frac{2}{2}\right)^2 = 1$:

$$x^2 + 2x + 1 - 1 = 3$$

Step 3: Rewrite the Left-Hand Side as a Perfect Square

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Now that we have added and subtracted the constant term, we can rewrite the left-hand side of the equation as a perfect square:

$$(x + 1)^2 - 1 = 3$$

Step 4: Simplify the Equation

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Finally, we can simplify the equation by adding 1 to both sides:

$$(x + 1)^2 = 4$$

Conclusion

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In this article, we have rewritten the equation $x^2 + 5x + \frac{5}{4} = 0$ in the form $(x-p)^2=q$. This form is known as the vertex form of a quadratic equation, where $p$ represents the x-coordinate of the vertex and $q$ represents the y-coordinate of the vertex. We have completed the square to obtain the vertex form, which involves adding and subtracting a constant term to the equation. This form is useful because it allows us to easily identify the vertex of the parabola represented by the quadratic equation. We have also provided example problems to demonstrate how to rewrite quadratic equations in the vertex form.