Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 5: Problem 27
Solve each equation by using the method of your choice. Find exact solutions. \(x^{2}-4 x+7=0\)
Short Answer
Expert verified
The solutions are \(x = 2 + i \sqrt{3}\) and \(x = 2 - i \sqrt{3}\).
Step by step solution
01
Identify the Equation Type
The equation given is a quadratic equation with the form \( ax^{2} + bx + c = 0 \), where \(a = 1\), \(b = -4\), and \(c = 7\). It doesn't factor nicely, so we'll use the quadratic formula.
02
Apply the Quadratic Formula
The quadratic formula is \( x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \). Here, \(a = 1\), \(b = -4\), and \(c = 7\). Plug these values into the formula.
03
Calculate the Discriminant
The discriminant is given by \( b^{2} - 4ac \). Substituting the values, we get \((-4)^{2} - 4 \times 1 \times 7 = 16 - 28 = -12\). Since the discriminant is negative, the solutions are complex.
04
Solve Using the Quadratic Formula
Substitute the discriminant back into the quadratic formula: \( x = \frac{4 \pm \sqrt{-12}}{2} \). Simplify to get \( x = \frac{4 \pm \sqrt{4 \times 3}i}{2} = \frac{4 \pm 2i \sqrt{3}}{2} \).
05
Simplify the Solutions
Simplify \( \frac{4 \pm 2i \sqrt{3}}{2} \) to get \( x = 2 \pm i \sqrt{3} \). Thus, the solutions are \( x = 2 + i \sqrt{3} \) and \( x = 2 - i \sqrt{3} \).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Formula
The quadratic formula is a powerful tool for finding the solutions of a quadratic equation. A quadratic equation has the standard form \( ax^2 + bx + c = 0 \). This formula is:
It tells us how to find the exact values of \( x \) that make the equation true, even when the solutions are not nice whole numbers.
Substituting the values of \( a \), \( b \), and \( c \) directly into the formula gives us the solutions in one step.
- \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
It tells us how to find the exact values of \( x \) that make the equation true, even when the solutions are not nice whole numbers.
Substituting the values of \( a \), \( b \), and \( c \) directly into the formula gives us the solutions in one step.
Discriminant
The discriminant plays a crucial role in determining the nature of the solutions of a quadratic equation. It is the part under the square root in the quadratic formula, given by:
- \( b^2 - 4ac \)
- If it is positive, there are two distinct real solutions.
- If it is zero, there is exactly one real solution.
- If it is negative, as in this exercise with \(-12\), it indicates that the solutions are complex numbers.
Complex Solutions
Complex solutions arise when the discriminant of a quadratic equation is negative. These solutions involve the imaginary unit \( i \), where \( i = \sqrt{-1} \). In our example, the discriminant was \(-12\), leading to solutions that include imaginary numbers.The quadratic formula will yield complex solutions in the form of:
For the equation \( x^2 - 4x + 7 = 0 \), the solutions were calculated as \( x = 2 \pm i\sqrt{3} \). These solutions mean that they do not lie on the real number line, but instead in the complex plane.
- \( x = p \pm qi \)
For the equation \( x^2 - 4x + 7 = 0 \), the solutions were calculated as \( x = 2 \pm i\sqrt{3} \). These solutions mean that they do not lie on the real number line, but instead in the complex plane.
Solving Equations
Solving quadratic equations involves finding the values of \( x \) that satisfy the equation. The method we used here was the quadratic formula, which is especially useful when equations cannot be easily factored.To solve:
- Identify \( a \), \( b \), and \( c \) from \( ax^2 + bx + c = 0 \).
- Compute the discriminant \( b^2 - 4ac \).
- Insert these into the quadratic formula.
- Solve the resulting equation, simplifying as necessary to obtain the solutions.
Polynomial Equations
Polynomial equations such as quadratics are those that involve sums of powers of variables. A quadratic is a second-degree polynomial, meaning the highest exponent is 2.
These types of equations can model real-world phenomena and provide foundational understanding for more complex mathematics.
Recognizing the form of polynomial equations helps in deciding which method to use for solving:
Recognizing the form of polynomial equations helps in deciding which method to use for solving:
- Factoring (when convenient and possible)
- Applying the quadratic formula
- Utilizing graphing techniques or numerical methods
