Question

Transform each of the following equations into another form. such that the new form of the equation contains no terms of the first degree. (1) \(x^{2}+y^{2}-4 x+10 y=0\) (2) \(x y+4 x-2 y+10=0\)

Short answer

Equation 1: \((x - 2)^2 + (y + 5)^2 = 29\) Equation 2: \(x = \frac{2(y+5)}{(y+4)}\)

Step-by-step solution

Equation 1: Completing the squares

To eliminate the first-degree terms in the given equation \(x^{2}+y^{2}-4x+10y=0\), we will complete the square for both x and y. Step 1: Group x and y terms together: \((x^2 - 4x) + (y^2 + 10y) = 0\) Step 2: Calculate half of the coefficients of the first-degree terms and square them: \((\frac{-4}{2})^2 = 4\) for x and \((\frac{10}{2})^2 = 25\) for y Step 3: Add and subtract these values to the equation: \((x^2 - 4x + 4) + (y^2 + 10y + 25) = 0 + 4 + 25\) Step 4: Factor the perfect square trinomials and simplify: \((x - 2)^2 + (y + 5)^2 = 29\) Now, the equation \((x - 2)^2 + (y + 5)^2 = 29\) has no terms of the first degree.

Equation 2: Factoring to eliminate first degree terms

To eliminate the first-degree terms in the given equation \(xy + 4x - 2y + 10 = 0\), we will factor out the first-degree terms. Step 1: Group full terms: \(x(y+4)-2(y+5)=0\) Step 2: Make a substitution: Let \(u=y+4\) and \(v=y+5\). The equation now becomes: \(xu-2v=0\) Step 3: Solve for x in terms of u and v: \(x = \frac{2v}{u}\) Step 4: Substitute the original expressions for u and v back into the equation: \(x = \frac{2(y+5)}{(y+4)}\) Now, the equation \(x = \frac{2(y+5)}{(y+4)}\) has no terms of the first degree.

Key concepts

Quadratic equations

Quadratic equations are polynomial equations of degree two, typically written in the form \(ax^2 + bx + c = 0\). They play a crucial role in algebra and geometry. The standard form contains:

• The quadratic term \(ax^2\)
• The linear term \(bx\)
• The constant term \(c\)

To solve for the values of \(x\), we can use:

• Factoring the equation, if possible
• The quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
• Completing the square, to transform the equation into a perfect square trinomial

Completing the square is a method used to make the quadratic equation look like \((x - p)^2 = q\). This transformation often simplifies solving or analyzing the properties of a quadratic, especially if dealing with conic sections.

Factoring

Factoring is a method used to simplify polynomials by expressing them as a product of their factors. When dealing with quadratic equations, factoring involves writing the quadratic expression as a product of two binomials.To factor an equation like \(ax^2 + bx + c = 0\), you need to find two numbers that multiply to \(ac\) and add to \(b\). These numbers help rewrite the middle term as a sum, thereby enabling factorization.In our exercise, factoring was applied differently to remove first-degree terms in \(xy + 4x - 2y + 10 = 0\):

• Terms were rearranged into two groups
• Substitutions were made to consolidate terms
• The equation simplified without first-degree terms

Factoring can often lead to elegant solutions, revealing the underlying structure of an expression and making further manipulations easier.

Coordinate transformation

Coordinate transformation refers to changing the coordinate system to simplify an equation or problem. In mathematics, it can involve translation, rotation, or scaling coordinates.For instance, completing the square effectively translates a quadratic equation by shifting the origin. This makes certain properties more apparent, like the center of a circle described by the equation \((x - 2)^2 + (y + 5)^2 = 29\).Steps in transforming coordinates:

• Identify the terms that need handling
• Adjust terms to form perfect squares or simplify expressions
• Interpret the transformed equation in its new form

Coordinate transformation is pivotal in fields like physics and engineering, allowing for more straightforward problem-solving by moving to a more convenient reference frame.