Problem 76
Question
Find all solutions of the quadratic equation. Relate the solutions of the equation to the zeros of an appropriate quadratic function. $$(x-2)^{2}=-16$$
Step-by-Step Solution
Verified Answer
The solutions of the quadratic equation are \(x = 2 + 4i\) and \(x = 2 - 4i\). These are also the zeros of the corresponding quadratic function \(f(x) = (x-2)^2 + 16 = 0\).
1Step 1: Simplify the equation
The original form of equation is \((x-2)^{2} = -16\). For simplicity, it needs to be rewritten in the standard quadratic equation form. In order to do this, the square root of both sides should be found, resulting in \(x-2 = \pm \sqrt{-16}\). Consequently, the equation becomes \(x-2 = \pm 4i\).
2Step 2: Solve for x
Now that the equation is simplified, solving for x becomes easier. Add 2 to both sides of the equation to isolate x. This results in two solutions: \(x = 2 + 4i\) and \(x = 2 - 4i\). These are the roots of the given quadratic equation.
3Step 3: Identify the zeros of the quadratic function
The solutions to the quadratic equation are also the zeros of the corresponding quadratic function. Hence, the zeros of the quadratic function \(f(x) = (x- (2 + 4i)) \cdot (x- (2 - 4i)) = (x-2)^2 + 16 = 0\) are \(2 + 4i\) and \(2 - 4i\).
Key Concepts
Complex NumbersQuadratic FunctionsRoots and ZerosImaginary Numbers
Complex Numbers
Complex numbers expand our understanding of real numbers by introducing a component called the imaginary unit, represented as \(i\). This imaginary unit is defined by the property \( i^2 = -1 \).
Complex numbers take the form \(a + bi\) where \(a\) and \(b\) are real numbers. Here, \(a\) represents the real part and \(bi\) represents the imaginary part. When dealing with quadratic equations like \((x-2)^2 = -16\), complex numbers become crucial.
In this equation, the conversion results in complex solutions due to the negative value inside the square root, specifically yielding \(x = 2 \pm 4i\).
These solutions illustrate how complex numbers are utilized even in seemingly straightforward quadratic equations, demonstrating their necessity in modern mathematics.
Complex numbers take the form \(a + bi\) where \(a\) and \(b\) are real numbers. Here, \(a\) represents the real part and \(bi\) represents the imaginary part. When dealing with quadratic equations like \((x-2)^2 = -16\), complex numbers become crucial.
In this equation, the conversion results in complex solutions due to the negative value inside the square root, specifically yielding \(x = 2 \pm 4i\).
These solutions illustrate how complex numbers are utilized even in seemingly straightforward quadratic equations, demonstrating their necessity in modern mathematics.
Quadratic Functions
Quadratic functions are mathematical expressions of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. They represent parabolas when graphed on a coordinate plane. In the given exercise, the quadratic function can be expressed as \((x-2)^2 + 16 = 0\).
Understanding quadratic functions involves:
Understanding quadratic functions involves:
- Recognizing the standard form \(ax^2 + bx + c\)
- Using the formula to find roots: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
- Relating solutions of the equation to the zeros of the function
Roots and Zeros
Roots or zeros of quadratic functions are the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). They are essentially where the graph of the function intersects the x-axis. In the example equation \((x-2)^{2} + 16 = 0\), by solving for \(x\), the roots found are \(2 + 4i\) and \(2 - 4i\).
These roots illustrate crucial concepts:
These roots illustrate crucial concepts:
- Each root can be complex, not just real numbers, involving imaginary parts.
- They verify the fundamental theorem of algebra, which states that a polynomial of degree \(n\) has exactly \(n\) roots in complex numbers.
Imaginary Numbers
Imaginary numbers, originated from square roots of negative numbers, provide a tool for solving equations that the real number system cannot accommodate. The imaginary unit, \(i\), is essential for defining these numbers, where \(i = \sqrt{-1}\).
Imaginary numbers appear prominently when handling quadratic equations with negative discriminants, such as in the \(x-2)^2 = -16\) case, leading us to the solutions \(x = 2 + 4i\) and \(x = 2 - 4i\).
These numbers broaden mathematical capabilities by:
Imaginary numbers appear prominently when handling quadratic equations with negative discriminants, such as in the \(x-2)^2 = -16\) case, leading us to the solutions \(x = 2 + 4i\) and \(x = 2 - 4i\).
These numbers broaden mathematical capabilities by:
- Allowing for the complete solution of quadratic equations.
- Facilitating calculations in fields like engineering and physics, especially in wave functions and signal processing.
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