Question

Rewrite each standard equation in general form. $$(x+3)^{2}+(y-5)^{2}=25$$

Short answer

The general form is \(x^2 + 6x + y^2 - 10y + 9 = 0\).

Step-by-step solution

Expand the Squared Binomials

First, you should expand the squared binomials using the identity \(a + b)^2 = a^2 + 2ab + b^2\). Apply this expansion to both \(x+3)^2\) and \(y-5)^2\).

Simplify the Equation

After expansion, simplify the equation by combining like terms. In this expansion, there will be no like terms, so simply write down the expanded form.

Write in General Form

To get the general form of the equation, \(Ax + By + C = 0\), you should get all terms on one side of the equation, leaving zero on the other side. This may involve subtracting the constant term from both sides of the expanded equation.

Key concepts

Quadratic Equations

Quadratic equations are a fundamental part of algebra and are characterized by an equation of the form \[ax^2 + bx + c = 0\] with a, b, and c being constants and with \( a eq 0 \). These equations can be solved for variable \( x \) and have a wide range of applications in various fields such as physics, engineering, and finance. The solutions to a quadratic equation are also known as the roots of the equation and can be found using various methods such as factoring, completing the square, using the quadratic formula, or graphing. In the context of this exercise, the equation of a circle \( (x+3)^2 + (y-5)^2 = 25 \) can be rearranged into a quadratic equation to represent it in general form, providing an insight into the properties of the circle such as its radius and center.

Expanding Binomials

The expansion of binomials is a crucial skill in algebra, which involves converting a binomial expression that has been raised to a power into a polynomial form. One commonly used identity for expanding binomials is \((a + b)^2 = a^2 + 2ab + b^2\), which is also known as the square of a binomial. In the exercise given, the process of expanding binomials allows us to simplify the equation of a circle from its standard form, \((x+3)^2 +(y-5)^2=25\), to an equation where each term is clearly visible. This exercise demonstrates the importance of being comfortable with the expansion of binomials to work with more complex algebraic expressions and in turning them into a format that is easier to manipulate for further analysis or graphing.

Circle Equations

The equations of circles in algebra are a specific application of quadratic equations. A circle can be represented in two major forms: standard form, \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center of the circle and \( r \) is its radius; and general form, \(Ax^2 + Ay^2 + Dx + Ey + F = 0\), which can be derived from the standard form. The conversion process showcases the relationship between the two forms and how the binomial expansion plays a role in morphing a geometrical shape into an algebraic equation. This altered equation still holds all the information about the circle's properties but displays it in a different arrangement that can be analyzed or used to find intersections with other algebraic expressions.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators that represent a specific value or set of values. They form the basis of algebra and are vital for expressing relationships and solving problems. In this exercise, the original equation of the circle is itself an algebraic expression in binomial form. Converting it to general form requires expanding and simplifying these binomials into a further developed algebraic expression that is equivalent to the original. This skill to transform and simplify expressions is crucial for understanding and solving more complex problems in algebra and other domains of mathematics.