Solving 10^(2x) + 11 = (x + 6)^2 - 2 Approximate Solutions
Finding solutions to complex equations often requires a blend of algebraic manipulation and numerical approximation techniques. In this article, we will delve into the process of solving the equation 10^(2x) + 11 = (x + 6)^2 - 2, identify the key steps involved, and pinpoint the approximate solutions from the given options. This mathematical exploration will highlight the practical applications of various mathematical principles and techniques.
Understanding the Equation 10^(2x) + 11 = (x + 6)^2 - 2
Before diving into the solution, let's break down the equation 10^(2x) + 11 = (x + 6)^2 - 2. This equation combines an exponential term (10^(2x)) with a quadratic term ((x + 6)^2 - 2), making it a transcendental equation. Transcendental equations are notoriously challenging to solve analytically, meaning there isn't a straightforward algebraic method to isolate x. Instead, we often rely on numerical methods or graphical techniques to find approximate solutions. The presence of both exponential and polynomial terms suggests that the solutions might not be integers or simple fractions, further necessitating the use of approximation methods.
Initial Simplification
To begin, let's simplify the equation to make it more manageable. We can start by expanding the quadratic term and rearranging the equation:
10^(2x) + 11 = (x + 6)^2 - 2 10^(2x) + 11 = x^2 + 12x + 36 - 2 10^(2x) + 11 = x^2 + 12x + 34
Now, let's move all the terms to one side to set the equation to zero:
10^(2x) - x^2 - 12x - 23 = 0
This form of the equation, 10^(2x) - x^2 - 12x - 23 = 0, is now ready for numerical methods or graphical analysis to find the solutions.
Graphical Approach to Finding Solutions
One effective method for approximating solutions to this equation is to use a graphical approach. We can graph two separate functions:
- f(x) = 10^(2x)
- g(x) = x^2 + 12x + 23
The solutions to the original equation are the x-coordinates of the points where these two graphs intersect. By plotting these functions, we can visually estimate the points of intersection and thus the approximate solutions. The exponential function f(x) = 10^(2x) will increase rapidly as x increases, while the quadratic function g(x) = x^2 + 12x + 23 will form a parabola. The intersections represent the values of x where the two functions are equal, satisfying our equation.
Analyzing the Behavior of the Functions
Before plotting, it's beneficial to analyze the behavior of each function. The exponential function f(x) = 10^(2x) is always positive and increases very sharply as x grows. For negative values of x, it approaches 0 but never reaches it. The quadratic function g(x) = x^2 + 12x + 23 is a parabola that opens upwards. We can find its vertex by completing the square or using the formula x = -b/(2a). In this case, the vertex is at x = -12/(21) = -6*. The y-coordinate of the vertex is g(-6) = (-6)^2 + 12(-6) + 23 = 36 - 72 + 23 = -13. This tells us the parabola has a minimum point at (-6, -13).
Numerical Methods for Approximation
While a graphical approach provides a visual understanding, numerical methods offer more precise approximations. One common method is the Newton-Raphson method, an iterative technique for finding the roots of a real-valued function. However, for the purpose of this article and the given options, we can use direct substitution to check which values are approximate solutions.
Direct Substitution Method
Given the options -9.6, -7.4, -4.6, -2.4, and 0.6, we can substitute each value into the simplified equation 10^(2x) - x^2 - 12x - 23 = 0 and see which ones make the equation approximately true. This method involves evaluating the left-hand side of the equation for each given x value and checking how close it is to zero. Let's evaluate each option:
-
x = -9.6 10^(2*(-9.6)) - (-9.6)^2 - 12(-9.6) - 23 ≈ 10^(-19.2) - 92.16 + 115.2 - 23 ≈ -0.000000000000000001 - 92.16 + 115.2 - 23 ≈ 0.039999999999999156
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x = -7.4 10^(2*(-7.4)) - (-7.4)^2 - 12(-7.4) - 23 ≈ 10^(-14.8) - 54.76 + 88.8 - 23 ≈ -54.76 + 88.8 - 23 ≈ 11.04
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x = -4.6 10^(2*(-4.6)) - (-4.6)^2 - 12(-4.6) - 23 ≈ 10^(-9.2) - 21.16 + 55.2 - 23 ≈ -21.16 + 55.2 - 23 ≈ 11.04
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x = -2.4 10^(2*(-2.4)) - (-2.4)^2 - 12(-2.4) - 23 ≈ 10^(-4.8) - 5.76 + 28.8 - 23 ≈ 0.00001584893192461113 - 5.76 + 28.8 - 23 ≈ 0.00001584893192461113
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x = 0.6 10^(2*(0.6)) - (0.6)^2 - 12(0.6) - 23 ≈ 10^(1.2) - 0.36 - 7.2 - 23 ≈ 15.848931924611127 - 0.36 - 7.2 - 23 ≈ -14.711068075388873
Identifying Approximate Solutions
From the direct substitution, we look for values that make the equation close to zero. The values -9.6 and -2.4 result in values closest to zero. Therefore, these are the approximate solutions to the equation.
Justification of the Solutions
- For x = -9.6, the result is approximately 0.04, which is very close to zero. This suggests that -9.6 is a good approximation for a solution.
- For x = -2.4, the result is approximately 0.00001584893192461113, which is extremely close to zero. This confirms -2.4 as a very accurate approximate solution.
Conclusion
Solving the equation 10^(2x) + 11 = (x + 6)^2 - 2 involves a combination of algebraic simplification and numerical approximation techniques. Through direct substitution, we have identified -9.6 and -2.4 as the approximate solutions. These solutions highlight the balance between exponential and polynomial behaviors in the equation. Understanding such equations is crucial in various fields, including engineering, physics, and computer science, where mathematical models often involve complex relationships between variables.
By using both graphical and numerical methods, we can effectively tackle transcendental equations and gain insights into their solutions. This exploration underscores the importance of a multifaceted approach to problem-solving in mathematics.
