Rewrite The Question 'Which Shows The Equation Below Written In The Form $ax^2 + Bx + C = 0$? $x + 9 = 4(x - 1)^2$' To 'Which Of The Following Equations Represents The Expression $x + 9 = 4(x - 1)^2$ In The Standard Quadratic Form $ax^2 + Bx + C = 0$?'
In the realm of mathematics, particularly algebra, the quadratic equation holds a position of paramount importance. It serves as a cornerstone for solving a myriad of problems across various disciplines, from physics and engineering to economics and computer science. At its core, a quadratic equation is a polynomial equation of the second degree, characterized by the presence of a term where the variable is raised to the power of two. The standard form of a quadratic equation is expressed as $ax^2 + bx + c = 0$, where 'a', 'b', and 'c' are constants, and 'x' represents the variable. The coefficient 'a' is of particular significance, as it cannot be zero; otherwise, the equation would degenerate into a linear equation.
The standard form of a quadratic equation is not merely a matter of convention; it is a powerful tool that facilitates the application of various solution techniques, such as factoring, completing the square, and the quadratic formula. Each of these methods relies on the specific arrangement of terms in the standard form to efficiently extract the roots or solutions of the equation. Understanding how to manipulate and transform equations into this standard form is, therefore, a fundamental skill for anyone venturing into the world of algebra and beyond.
In this article, we delve into the process of converting a given equation into the standard quadratic form. We will dissect the steps involved, unraveling the algebraic manipulations required to achieve the desired format. By mastering this skill, you will unlock a gateway to solving a wide array of mathematical problems and gain a deeper appreciation for the elegance and power of quadratic equations.
Transforming $x + 9 = 4(x - 1)^2$ into Standard Form
The question at hand presents us with an equation, $x + 9 = 4(x - 1)^2$, and challenges us to express it in the standard quadratic form, which is $ax^2 + bx + c = 0$. This task requires us to navigate through a series of algebraic manipulations, carefully expanding, simplifying, and rearranging terms until we arrive at the desired format. Let's embark on this journey, step by step, to unravel the transformation process.
Our initial equation, $x + 9 = 4(x - 1)^2$, presents a structure that deviates from the standard quadratic form. The right-hand side of the equation contains a squared term, which needs to be expanded to reveal the quadratic nature of the equation. The first step, therefore, involves expanding the term $(x - 1)^2$. This can be achieved by applying the binomial expansion formula or by simply multiplying the expression by itself: $(x - 1)(x - 1)$. Either approach will yield the same result: $x^2 - 2x + 1$.
Now, we substitute this expanded form back into our original equation, giving us $x + 9 = 4(x^2 - 2x + 1)$. The next step is to distribute the constant '4' across the terms within the parentheses on the right-hand side. This involves multiplying each term inside the parentheses by 4, resulting in the equation $x + 9 = 4x^2 - 8x + 4$. We are now one step closer to achieving the standard quadratic form. The equation now exhibits a quadratic term ($4x^2$), a linear term ($-8x$), and a constant term (4) on the right-hand side, while the left-hand side contains a linear term ($x$) and a constant term (9).
To bring the equation into the standard form $ax^2 + bx + c = 0$, we need to consolidate all terms on one side of the equation, leaving zero on the other side. The most common approach is to move all terms to the side containing the quadratic term, which in this case is the right-hand side. To accomplish this, we subtract $x$ and 9 from both sides of the equation. This operation ensures that the equation remains balanced while simultaneously shifting the terms to the desired location.
Subtracting $x$ from both sides, we get $9 = 4x^2 - 9x + 4$. Next, subtracting 9 from both sides, we arrive at $0 = 4x^2 - 9x - 5$. This equation is now in the standard quadratic form $ax^2 + bx + c = 0$, where $a = 4$, $b = -9$, and $c = -5$. We have successfully transformed the original equation into its standard quadratic form, paving the way for the application of various solution techniques.
Identifying the Correct Standard Form Equation
Having successfully transformed the given equation, $x + 9 = 4(x - 1)^2$, into the standard quadratic form, we now stand equipped to identify the correct option from the provided choices. The transformed equation, as we derived, is $4x^2 - 9x - 5 = 0$. This equation represents the standard form, where the coefficients 'a', 'b', and 'c' correspond to 4, -9, and -5, respectively. The structure of the equation perfectly aligns with the general form $ax^2 + bx + c = 0$.
Now, let's scrutinize the given options to pinpoint the one that matches our derived equation. Option A presents the equation $4x^2 - 9x + 13 = 0$. While this equation shares the same quadratic and linear terms as our derived equation, the constant term differs. In Option A, the constant term is +13, whereas in our derived equation, it is -5. Therefore, Option A does not represent the correct standard form of the given equation.
Moving on to Option B, we encounter the equation $4x^2 - 7x - 5 = 0$. This equation also exhibits a discrepancy when compared to our derived equation. The linear term in Option B is -7x, while in our derived equation, it is -9x. Although the quadratic and constant terms match, the difference in the linear term disqualifies Option B as the correct answer.
Option C presents the equation $4x^2 - 7x + 13 = 0$. This option deviates from our derived equation in both the linear and constant terms. The linear term is -7x, and the constant term is +13, neither of which aligns with the corresponding terms in our derived equation. Consequently, Option C cannot be the correct standard form representation.
Finally, we arrive at Option D, which presents the equation $4x^2 - 9x - 5 = 0$. A careful comparison of this equation with our derived equation reveals a perfect match. The quadratic term, linear term, and constant term all align seamlessly. Therefore, Option D stands as the correct answer, accurately representing the standard quadratic form of the given equation.
In conclusion, the journey of transforming the equation $x + 9 = 4(x - 1)^2$ into its standard quadratic form has not only provided us with the solution but has also reinforced the importance of meticulous algebraic manipulation and the significance of the standard form in solving quadratic equations. The correct answer, as we have identified, is Option D: $4x^2 - 9x - 5 = 0$.
Importance of Standard Quadratic Form
The standard quadratic form, expressed as $ax^2 + bx + c = 0$, is more than just a conventional way of writing quadratic equations; it is a fundamental structure that unlocks the gateway to a plethora of solution techniques and analytical insights. The arrangement of terms in this specific format is not arbitrary; it is carefully designed to facilitate the application of various methods for finding the roots or solutions of the equation. Understanding the significance of the standard form is crucial for anyone seeking to master the art of solving quadratic equations and harnessing their power in diverse mathematical and real-world applications.
One of the primary reasons why the standard form is so important lies in its compatibility with the quadratic formula. The quadratic formula, a cornerstone of algebra, provides a universal method for solving any quadratic equation, regardless of its complexity. The formula, given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, directly utilizes the coefficients 'a', 'b', and 'c' from the standard form. Without the equation being in standard form, the direct application of the quadratic formula would be cumbersome, if not impossible. The standard form, therefore, serves as the key that unlocks the power of the quadratic formula, enabling us to efficiently find the solutions of any quadratic equation.
Another significant advantage of the standard form is its role in the factoring method. Factoring is a technique that involves expressing the quadratic equation as a product of two linear factors. This method is particularly efficient when the roots of the equation are rational numbers. The standard form provides a clear framework for identifying potential factors and systematically decomposing the equation. The coefficients 'a', 'b', and 'c' in the standard form guide the factoring process, helping us to determine the appropriate combination of factors that will yield the original equation. Without the standard form, the factoring process would be significantly more challenging and less intuitive.
Furthermore, the standard form plays a crucial role in the method of completing the square. Completing the square is a technique that involves manipulating the quadratic equation to create a perfect square trinomial on one side. This method is particularly useful when the equation cannot be easily factored. The standard form provides the necessary foundation for completing the square. By rearranging the terms and adding a specific constant to both sides of the equation, we can transform the equation into a form where one side is a perfect square, making it easier to solve. The coefficients 'a', 'b', and 'c' in the standard form guide the process of determining the constant needed to complete the square.
Beyond solution techniques, the standard form also provides valuable insights into the nature of the quadratic equation itself. The coefficients 'a', 'b', and 'c' reveal important information about the graph of the quadratic function represented by the equation. The coefficient 'a' determines the concavity of the parabola, indicating whether it opens upwards or downwards. The coefficient 'b' influences the position of the axis of symmetry, while the coefficient 'c' represents the y-intercept of the parabola. By analyzing these coefficients in the standard form, we can gain a deeper understanding of the behavior and characteristics of the quadratic equation.
In conclusion, the standard quadratic form is not merely a notational preference; it is a powerful tool that facilitates the solution of quadratic equations and provides valuable insights into their nature. Its compatibility with various solution techniques, such as the quadratic formula, factoring, and completing the square, makes it an indispensable asset in the realm of algebra and beyond. Understanding the importance of the standard form is essential for anyone seeking to master the art of solving quadratic equations and harnessing their power in diverse mathematical and real-world applications.
In summary, transforming equations into the standard quadratic form $ax^2 + bx + c = 0$ is a fundamental skill in algebra. This form allows for the easy application of methods like the quadratic formula, factoring, and completing the square. In the specific case of the equation $x + 9 = 4(x - 1)^2$, we demonstrated the steps to expand, simplify, and rearrange the equation to match the standard form. The correct transformation yields $4x^2 - 9x - 5 = 0$, which corresponds to option D. This exercise highlights the importance of algebraic manipulation and the utility of the standard quadratic form in solving mathematical problems.