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Magazine Chapter 5: Problem 15
Solve each equation using the Quadratic Formula. $$ x^{2}+3 x+5=0 $$
Short Answer
Expert verified
The solutions to the equation are \( x = -\frac{3}{2} + \frac{\sqrt{11}}{2}i \) and \( x = -\frac{3}{2} - \frac{\sqrt{11}}{2}i \).
Step by step solution
01
Identify the coefficients
For the equation in the form of a quadratic equation, which is typically written as \( ax^2 + bx + c = 0 \), identify the coefficients of \( x^2 \), \( x \), and the constant term. In this case, \( a=1 \), \( b=3 \), and \( c=5 \).
02
State the Quadratic Formula
The Quadratic Formula is used to find the solutions to a quadratic equation of the form \( ax^2 + bx + c = 0 \). The formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). We will use this to find the roots of the equation.
03
Substitute the coefficients into the Quadratic Formula
Substitute \( a=1 \), \( b=3 \), and \( c=5 \) into the Quadratic Formula: \( x = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 1 \cdot 5}}{2 \cdot 1} \).
04
Simplify under the square root
Simplify the expression under the square root (the discriminant): \( 3^2 - 4 \cdot 1 \cdot 5 = 9 - 20 = -11 \). Since the discriminant is negative, this tells us that the solutions will be complex numbers.
05
Compute the solutions
Compute the solutions using the square root of the negative discriminant: \( x = \frac{-3 \pm \sqrt{-11}}{2} \). We can express the square root of a negative number using \( i \) (the imaginary unit), where \( i^2 = -1 \). Therefore, the solutions are \( x = \frac{-3 \pm \sqrt{11}i}{2} \).
06
Simplify the solutions
Write the final simplified solutions: \( x = -\frac{3}{2} \pm \frac{\sqrt{11}}{2}i \). There are two complex solutions to the equation.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Solving Quadratic Equations
Solving quadratic equations is a fundamental task in algebra. A quadratic equation takes the general form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( a \) is not zero. The most famous method to solve these equations is the Quadratic Formula: \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \).
To use the formula effectively, follow these steps:
To use the formula effectively, follow these steps:
- Identify \( a \), \( b \), and \( c \) from your equation. In your example, \( a=1 \), \( b=3 \), and \( c=5 \).
- Plug these values into the Quadratic Formula.
- Simplify the expression to find your solutions, \( x \).
Complex Numbers
Complex numbers are an extension of the real numbers and are important in understanding solutions to certain quadratic equations. They have the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit that satisfies \( i^2 = -1 \).
When the discriminant (the part of the Quadratic Formula under the square root) is negative, like \( -11 \) in your example, the solutions involve complex numbers:
When the discriminant (the part of the Quadratic Formula under the square root) is negative, like \( -11 \) in your example, the solutions involve complex numbers:
- The square root of a negative number cannot be simplified using real numbers alone. Hence, complex numbers come into play.
- Express the square root of the negative number using \( i \) to facilitate further calculation.
Discriminant
The discriminant in a quadratic equation is found under the square root in the Quadratic Formula and is key in determining the nature of the solutions. It is given by \( b^2 - 4ac \).
The value of the discriminant reveals the following:
The value of the discriminant reveals the following:
- If the discriminant is positive, there are two distinct real solutions.
- If the discriminant is zero, there is exactly one real solution (also called a repeated or double root).
- If the discriminant is negative, there are no real solutions, but two complex solutions.
Imaginary Unit
The imaginary unit, denoted by \( i \), is a mathematical concept used for the square root of \( -1 \). It is essential for working with complex numbers where the need to take the square root of a negative number arises. In your quadratic equation, when the discriminant turned out to be negative, the imaginary unit made it possible to find meaningful solutions.
The properties of \( i \) are the foundation of complex arithmetic:
The properties of \( i \) are the foundation of complex arithmetic:
- \( i^2 = -1 \)
- \( i^3 = -i \)
- \( i^4 = 1 \), and it repeats cyclically from there.
