Standard Form Of Quadratic Equation 9 - 7x = (4x - 3)^2 + 5

Let's embark on a journey to transform the given equation into its standard form. Understanding the standard form of a quadratic equation is crucial for various mathematical operations, including solving for roots, graphing parabolas, and analyzing the equation's properties. This comprehensive guide will walk you through each step, ensuring a clear grasp of the process. The original equation we're tackling is:

$$9 - 7x = (4x - 3)^2 + 5$$

Our mission is to rewrite this equation in the standard form, which is generally expressed as:

$$ax^2 + bx + c = 0$$

where a, b, and c are constants, and x is the variable. Let's dive into the step-by-step process.

Step 1: Expanding the Squared Term

The first key step in converting this equation to standard form involves expanding the squared term, $(4x - 3)^2$. Remember that squaring a binomial means multiplying it by itself. We'll apply the FOIL (First, Outer, Inner, Last) method or the binomial theorem to achieve this expansion accurately. This initial expansion is paramount as it sets the stage for combining like terms and simplifying the equation into the desired quadratic form. Neglecting this step or performing it incorrectly can lead to a cascade of errors in subsequent steps. Therefore, meticulous attention to detail is crucial at this stage. We have:

$$(4x - 3)^2 = (4x - 3)(4x - 3)$$

Applying the FOIL method:

• First: $(4x)(4x) = 16x^2$
• Outer: $(4x)(-3) = -12x$
• Inner: $(-3)(4x) = -12x$
• Last: $(-3)(-3) = 9$

Combining these terms, we get:

$$16x^2 - 12x - 12x + 9$$

Simplifying further:

$$16x^2 - 24x + 9$$

This expansion is a critical component of the transformation, paving the way for the subsequent steps in achieving the standard quadratic form. Now, let’s substitute this expansion back into the original equation.

Step 2: Substituting the Expanded Term Back into the Equation

Now that we've successfully expanded the squared term, the next crucial step is to substitute it back into the original equation. This substitution is a pivotal moment in our transformation process as it brings us closer to the standard quadratic form. It's akin to replacing a piece in a puzzle, and this replacement is essential for the overall picture to emerge clearly. The accuracy of this substitution is paramount; any error here will propagate through the remaining steps, potentially leading to an incorrect final answer. Therefore, double-checking the substitution is a recommended practice to ensure precision. Our original equation was:

$$9 - 7x = (4x - 3)^2 + 5$$

Substituting the expanded form, $16x^2 - 24x + 9$, for $(4x - 3)^2$, we get:

$$9 - 7x = 16x^2 - 24x + 9 + 5$$

This substitution is more than just a mechanical step; it's a strategic move that allows us to consolidate the terms and rearrange the equation. By replacing the squared term with its expanded equivalent, we've effectively laid the foundation for the next phase of simplification. The equation now contains individual terms that can be combined and rearranged, paving the way for isolating all terms on one side and setting the equation equal to zero. This process of substitution is a cornerstone of algebraic manipulation, and its proper execution is vital for achieving the desired outcome. With the expanded term now in place, we can proceed to the next step: combining like terms and rearranging the equation.

Step 3: Combining Like Terms and Rearranging the Equation

With the expanded term substituted, our next objective is to combine like terms and rearrange the equation. This step is akin to organizing the elements of a mathematical expression, bringing together similar components to simplify the overall structure. It’s a crucial stage in the process of converting the equation to standard form, as it consolidates the various terms and prepares the equation for the final rearrangement. Accuracy in this step is paramount; any misidentification or incorrect combination of like terms can lead to errors that affect the final result. Let's revisit the equation we have:

$$9 - 7x = 16x^2 - 24x + 9 + 5$$

First, let’s combine the constant terms on the right side of the equation:

$$9 + 5 = 14$$

So, the equation becomes:

$$9 - 7x = 16x^2 - 24x + 14$$

Now, our goal is to set the equation to zero, which is a hallmark of the standard quadratic form. To achieve this, we'll move all the terms from the left side of the equation to the right side. This process involves performing inverse operations to effectively cancel out terms on the left and transfer them to the right. We'll add $7x$ to both sides:

$$9 - 7x + 7x = 16x^2 - 24x + 14 + 7x$$

This simplifies to:

$$9 = 16x^2 - 17x + 14$$

Next, we'll subtract 9 from both sides:

$$9 - 9 = 16x^2 - 17x + 14 - 9$$

This results in:

$$0 = 16x^2 - 17x + 5$$

By combining like terms and strategically rearranging the equation, we've successfully transformed it into a form that closely resembles the standard quadratic form. The equation is now poised for the final touch: ensuring it's presented in the conventional standard form arrangement. This meticulous process of combining and rearranging terms is a testament to the structured approach required in algebraic manipulations.

Step 4: Expressing the Equation in Standard Form

Now that we've combined like terms and rearranged the equation, the final step is to express it precisely in the standard form of a quadratic equation, which is: $ax^2 + bx + c = 0$. This standard form is not just a matter of aesthetics; it's a convention that facilitates various mathematical operations, from solving the equation to analyzing the properties of the corresponding parabola. Ensuring the equation is in this specific form makes it readily accessible for further analysis and manipulation. The equation we've arrived at is:

$$0 = 16x^2 - 17x + 5$$

To strictly adhere to the standard form, we can simply rewrite the equation by swapping the sides. While mathematically equivalent, presenting it with the quadratic expression on the left side and zero on the right side aligns perfectly with the conventional standard form representation. This seemingly minor adjustment is a significant step in ensuring clarity and consistency in mathematical communication. Rewriting the equation, we get:

$$16x^2 - 17x + 5 = 0$$

This is the standard form of the given equation. It clearly presents the quadratic, linear, and constant terms in their respective positions, making it easy to identify the coefficients a, b, and c. In this case, a = 16, b = -17, and c = 5. Expressing the equation in this standard form is the culmination of our step-by-step transformation process. It represents the final, polished version of the equation, ready for any subsequent mathematical analysis or problem-solving techniques. The journey from the original equation to this standard form highlights the power of algebraic manipulation and the importance of adhering to mathematical conventions. By meticulously following each step, we've not only arrived at the solution but also reinforced the understanding of quadratic equations and their standard form representation.

Conclusion

Therefore, the equation $9 - 7x = (4x - 3)^2 + 5$ when written in standard form is: $16x^2 - 17x + 5 = 0$, which corresponds to option C. This step-by-step solution demonstrates the process of expanding, simplifying, and rearranging an equation to fit the standard quadratic form, a fundamental skill in algebra.