The given equation appears to be a quadratic equation in the form of ax^2 + bx + c. However, it seems there might be a typo or confusion in the representation of the equation. For the purpose of providing a clear and authoritative solution, let's assume the equation is intended to be x^2 + 2x + 3x + 1.
Simplifying and Solving the Quadratic Equation
To simplify and solve the equation, we first need to combine like terms:
x^2 + 2x + 3x + 1 = x^2 + 5x + 1
This is a standard form of a quadratic equation: ax^2 + bx + c = 0, where a = 1, b = 5, and c = 1.
Applying the Quadratic Formula
The solutions to a quadratic equation can be found using the quadratic formula:
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
Substituting a = 1, b = 5, and c = 1 into the formula:
x = \frac{-5 \pm \sqrt{5^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}
x = \frac{-5 \pm \sqrt{25 - 4}}{2}
x = \frac{-5 \pm \sqrt{21}}{2}
| Solution | Value |
|---|---|
| Solution 1 | \frac{-5 + \sqrt{21}}{2} |
| Solution 2 | \frac{-5 - \sqrt{21}}{2} |
Key Points
- The equation simplifies to x^2 + 5x + 1.
- The solutions are given by the quadratic formula.
- The discriminant b^2 - 4ac = 21, indicating real and distinct roots.
- The solutions are \frac{-5 + \sqrt{21}}{2} and \frac{-5 - \sqrt{21}}{2}.
- The equation does not factor easily, requiring the use of the quadratic formula.
Contextual Interpretation and Practical Application
Quadratic equations like x^2 + 5x + 1 = 0 appear in various contexts, including physics, engineering, and economics. They are used to model projectile motion, optimize functions, and solve problems involving area and volume.
Historical Context and Evolutionary Developments
The quadratic formula has its roots in ancient civilizations, with contributions from mathematicians such as Al-Khwarizmi. Over time, the formula has been refined and applied in numerous fields, demonstrating its versatility and significance.
What is the quadratic formula?
+The quadratic formula is given by x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, which provides the solutions to a quadratic equation of the form ax^2 + bx + c = 0.
How do you determine the nature of the roots?
+The nature of the roots can be determined by the discriminant b^2 - 4ac. If it is positive, the roots are real and distinct. If it is zero, the roots are real and equal. If it is negative, the roots are complex.
Can the quadratic formula be applied to all types of equations?
+The quadratic formula is specifically designed for quadratic equations of the form ax^2 + bx + c = 0. It may not be directly applicable to other types of equations, such as linear, cubic, or higher-order polynomial equations.
In conclusion, the solutions to the equation (x^2 + 5x + 1 = 0) are (\frac{-5 + \sqrt{21}}{2}) and (\frac{-5 - \sqrt{21}}{2}), demonstrating the application of the quadratic formula in solving quadratic equations.