Problem 33 - Algebra 2 and Trigonometry

Question

In \(18-35,\) find each common solution algebraically. Express irrational roots in simplest radical form. $$ \begin{array}{l}{x^{2}+y^{2}=24} \\ {x+y=8}\end{array} $$

Step-by-Step Solution

Answer

The solutions are \((4+2i, 4-2i)\) and \((4-2i, 4+2i)\).

Step 1: Express One Variable in Terms of Another

First, solve the equation \(x + y = 8\) for one variable in terms of the other. Let's express \(y\) in terms of \(x\):\[ y = 8 - x \]

Step 2: Substitute and Form a Quadratic Equation

Substitute \(y = 8 - x\) into the first equation \(x^2 + y^2 = 24\):\[ x^2 + (8 - x)^2 = 24 \]

Step 3: Expand and Simplify the Equation

Expand \((8 - x)^2\):\[ x^2 + (64 - 16x + x^2) = 24 \]Combine like terms:\[ 2x^2 - 16x + 64 = 24 \]

Step 4: Rearrange the Equation

Subtract 24 from both sides to simplify:\[ 2x^2 - 16x + 40 = 0 \]Divide every term by 2 to make it easier to solve:\[ x^2 - 8x + 20 = 0 \]

Step 5: Solve the Quadratic Equation Using the Quadratic Formula

Use the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = -8\), and \(c = 20\).Calculate the discriminant:\[ b^2 - 4ac = (-8)^2 - 4 \cdot 1 \cdot 20 = 64 - 80 = -16 \]Since the discriminant is negative, the roots are complex.

Step 6: Find the Complex Roots

The quadratic formula gives:\[ x = \frac{8 \pm \sqrt{-16}}{2} \]Since \(\sqrt{-16} = 4i\), substitute it in:\[ x = 4 \pm 2i \]

Step 7: Determine Corresponding y Values

From \(y = 8 - x\), calculate \(y\) for each \(x\):For \(x = 4 + 2i\), \(y = 8 - (4 + 2i) = 4 - 2i\).For \(x = 4 - 2i\), \(y = 8 - (4 - 2i) = 4 + 2i\).

Step 8: Summarize the Solutions

The solutions to the system are the pairs of complex numbers:\((x, y) = (4 + 2i, 4 - 2i)\) and \((4 - 2i, 4 + 2i)\).

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation of the second degree. It typically has the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). The highest degree of this equation is 2, because of the \( x^2 \) term. This type of equation can often describe various physical phenomena, from the trajectories of projectiles to optimization problems.

In our problem, we start with the equation \( x^2 - 8x + 20 = 0 \), which is derived from the initial conditions given by substituting one equation into another. Understanding quadratic equations is crucial, as they lay the foundation for solving more complex expressions, such as those involving complex numbers.

Complex Numbers

Complex numbers are an extension of the real numbers and are used to solve equations that do not have real solutions. A complex number is expressed in the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit with the property \( i^2 = -1 \).

In the exercise, when the quadratic equation resulted in a negative discriminant, it implied that the solutions would be complex. The roots of the equation \( x^2 - 8x + 20 = 0 \) are complex, specifically \( 4 + 2i \) and \( 4 - 2i \). Complex solutions frequently appear in physics, engineering, and other advanced mathematics fields.

Quadratic Formula

The quadratic formula is a universal method for solving any quadratic equation. It is expressed as \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). With this formula, you can find the solutions of a quadratic equation by plugging in the values of \( a \), \( b \), and \( c \).

In our exercise, using \( a = 1 \), \( b = -8 \), and \( c = 20 \), the quadratic formula gives:

• \( x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4 \cdot 1 \cdot 20}}{2 \cdot 1} \)
• \( x = \frac{8 \pm \sqrt{-16}}{2} \)

Because the discriminant is negative, the solutions are complex numbers. This illustrates the quadratic formula's flexibility in finding complex solutions.

Discriminant

The discriminant is a part of the quadratic formula expressed as \( D = b^2 - 4ac \). It gives important information about the nature of the roots of a quadratic equation.

If the discriminant is:

• Positive: The quadratic equation has two distinct real roots.
• Zero: The quadratic equation has exactly one real root, which means the roots are identical (or repeated).
• Negative: The quadratic equation has two complex roots.

In this exercise, the discriminant \( b^2 - 4ac \) equals \( -16 \), indicating that there are complex solutions. This underscores the power of understanding the discriminant, as it quickly tells us the nature of the solutions.