Question

Rewrite standard equation in general form. $$(y-5)^{2}=9(x+8)$$

Short answer

The general form of the given standard equation is \( -9x + 9y^2 - 90y + 153 = 0\).

Step-by-step solution

Expand the Square

First, we expand the square on the left side of the equation. Using the identity \(a - b)^2 = a^2 - 2ab + b^2\), the expansion of \(y-5)^2\) gives \(y^2 - 10y + 25\).

Distribute the Right Side

Multiply both sides of the equation by 9 to get rid of the coefficient on the right side: \(9(y^2 - 10y + 25) = 9(x + 8)\). This yields \(9y^2 - 90y + 225 = 9x + 72\).

Rearrange in the General Form

We want to write the equation in the form \(Ax + By + C = 0\). To do this, we subtract \(9x + 72\) from both sides to get: \(9y^2 - 90y + 225 - 9x - 72 = 0\).

Combine Like Terms

Combining like terms on the left side results in the equation \(9y^2 - 90y + 153 - 9x = 0\).

Rearrange Terms

Finally, for aesthetic reasons, we can rearrange the terms to put the \(x\) term first: \( -9x + 9y^2 - 90y + 153 = 0\). Normally, the \(x\) term is written first, and since the general form is \(Ax + By + C = 0\), we can also write the equation as \( -9x + 9y^2 - 90y + 153 = 0\).

Key concepts

Expanding the Square

To convert a quadratic equation from standard form to general form, one key step is expanding the square. This process involves applying the algebraic identity \(a^2 - 2ab + b^2\), which represents the expansion of \(a - b)^2\). In the context of our exercise, the square that needs to be expanded is \(y-5)^2\).

When we expand \(y-5)^2\), we compute each term according to the identity: \(y^2\) for the \(a^2\) part, \(2 \cdot 5 \cdot y\) or \(10y\) for the \(2ab\) part, and \(5^2\) or 25 for the \(b^2\) part. These expanded terms \(y^2 - 10y + 25\) replace the original squared term in the equation, paving the way for further rearrangement into the general form of the equation.

General Form of a Conic Section

A conic section can be represented in various forms, with one common representation being the general form. It's a single equation, typically arranged as \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\), where \(A\), \(B\), \(C\), \(D\), \(E\), and \(F\) are coefficients that can take on any value, making it possible to express circles, ellipses, parabolas, and hyperbolas.

In our exercise, we deal with a parabola. The given equation does not have an \(xy\) term or an \(x^2\) term, indicating that it's aligned with the coordinate axes. The process of rewriting it into the general form involves distributing, rearranging, and combining like terms until no parentheses remain, and the equation is in the form of \(Ax + By + C = 0\). This form is extremely versatile and is commonly used for graphing, analyzing, and solving problems involving conic sections.

Rearranging Algebraic Equations

One essential aspect of algebra is the rearrangement of equations. This skill is vital for manipulating an equation into a desired form. In the context of our exercise, we first expanded the square and distributed the right side. The next step was to rewrite the quadratic equation in the general form. This was accomplished by moving all terms to one side of the equation and combining like terms.

To master this skill, one must understand how to balance equations by performing the same operation on both sides, the ability to combine like terms, and the knowledge of when to rearrange terms to achieve a standard format, such as \(Ax + By + C = 0\). By doing this carefully, we avoid mistakes and ensure that the equation remains equivalent throughout the process. In our final step, we arranged the terms in descending order of degree and ensured that the \(x\) term was listed first, mirroring the standard \(Ax + By + C\) structure of the general form.