Problem 127 - A Graphical Approach to College Algebra

Question

Solve each equation for \(x\) and then for \(y\). $$x^{2}+x y+y^{2}=0 \quad(x>0, y>0)$$

Step-by-Step Solution

Answer

Equation needs complex solutions, real for both x and y is not attainable.

Step 1: Substitute y in terms of x

Given the equation \(x^{2}+ x y + y^{2} = 0\), let's express \(y\) in terms of \(x\). We can factor the equation as \((x + y)^2 - xy = 0\). Then, \(xy = (x + y)^2\). Substitute \(y = -\frac{x}{2} + i\frac{\sqrt{3}}{2}x\).

Step 2: Real and Imaginary Components

From \(y = -\frac{x}{2} + i\frac{\sqrt{3}}{2}x\), we note that for real values as asked, we extract that \(x/y\) must be a complex conjugate leading to elimination of imaginary part. Thus, both parts separately must equate to zero for real solutions, meaning \(x = y\).

Step 3: Solve for x and y

Substituting back the solution \(x = y\) into the original equation gives: \(x^2 + x^2 + x^2 = 0\). This simplifies to \(3x^2 = 0\). Therefore, \(x = 0\), which contradicts the condition \(x > 0\). Thus, re-evaluating complex solutions is valid for x and positive y.

Step 4: Verify equations for positive solutions

For the given condition \(x > 0\), \(y > 0\), check values associated to prior complex form deriving exact conditions for solution or rewrite equations.

Step 5: Conclusion

Upon further re-evaluation, positive solution yielding in this form doesn't hold, is rather conventionally used for hypothetical transformation enabling scaling beyond real numbers.

Key Concepts

Quadratic Equations

Quadratic equations are equations of the form \( ax^2 + bx + c = 0 \) where \( a \), \( b \), and \( c \) are constants, and \( x \) represents an unknown variable. Solving quadratic equations gives you values for \( x \) that make the equation true. These equations are fundamental in algebra as they model a wide range of phenomena in science and engineering.
There are various methods to solve quadratic equations:

• Factoring: When the quadratic can be expressed as a product of two binomials.
• The Quadratic Formula: \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \), which can solve any quadratic equation.
• Completing the Square: Rewriting the equation to make it easier to solve.

In the given exercise, the equation \(x^2 + xy + y^2 = 0\) is manipulated using transformations related to the quadratic form, but solution in complex domain indicates scenarios extending beyond simple quadratic solutions, especially when dealing with complex numbers.

Imaginary Numbers

Imaginary numbers arise when we have to take the square root of a negative number. The simplest imaginary number is \(i\), defined as \( \sqrt{-1} \). Thus, any imaginary number can be expressed as \( bi \), where \( b \) is a real number.
This concept extends the number system to complex numbers, which are composed of a real part and an imaginary part, noted as \( a + bi \). Imaginary numbers are crucial since they enable the solution of equations where traditional real numbers are insufficient.
In the given exercise, introducing \( y = -\frac{x}{2} + i\frac{\sqrt{3}}{2}x \) suggests using complex numbers due to non-zero imaginary component indicating complex conjugate pair use for simplifying purposes. Imaginary numbers thus offer a broader perspective where equations involving non-real solutions like this can be fulfilled.

Complex Conjugates

Complex conjugates are pairs of complex numbers that have the same real part but opposite imaginary parts. If you have a complex number \( a + bi \), its complex conjugate is \( a - bi \).
Complex conjugates are valuable because when you multiply a complex number by its conjugate, the result is a real number: \((a+bi)(a-bi) = a^2 + b^2\). This property significantly aids in simplifying expressions and solving equations involving complex numbers.
In the homework solution, the idea of using complex conjugates appears when addressing symmetry in imaginary numbers to ensure the equation's real component can become zero, balancing out the equation. This underscores the usage of conjugates to provide clarity in solving complex quadratic forms as seen in practice exercises like the presented one.