91Ó°ÊÓ

A quadratic equation is any equation that can be rewritten in the form \[ ax^2 + bx + c = 0 \] where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). In our problem, the first equation is similar to this form if you rearrange it: \[ x = 25 - y^2 \]Even though it is not directly in the standard form, we can see that we're working with quadratic terms due to the \(y^2\) component. To find the solutions for the variable \(y\), we arranged the equation in a way that isolates \(x\) and makes it easier to substitute into another equation.

• The equation essentially describes a parabola when graphed in the \(xy\)-plane.
• Quadratic equations often have two solutions, which in cases like squares, can represent distinct or repeated values.

By substituting these solutions into another equation, like in a system of equations, you can find corresponding values for other variables, like \(x\) in this instance.

Exponential functions are expressions in which a variable appears in the exponent, typically formatted as \[ f(x) = a \cdot b^x \] where \(a\) and \(b\) are constants and \(b > 0\). In our problem, the second equation is an exponential equation: \[ e^{3xy} = 1 \] Here, \(e\) is the base of the natural logarithms, approximately equal to 2.718. To solve this type of equation, we often convert it into a logarithmic form.

• Exponential functions are characterized by their rapid growth or decay.
• They are used extensively in different scientific fields to model growth processes, such as population growth or radioactive decay.

This equation's specific form simplifies when equated to 1, as any number raised to the zero power is 1. Thus, our primary task becomes finding when the exponent evaluates to zero.

The logarithmic form is the inverse process of exponential expressions. It allows us to solve equations where exponents are unknown. For an exponential equation like \[ a^x = N \] we can express it in logarithmic form as \[ x = \log_a(N) \]. In the context of our problem, where the equation is \[ e^{75 - 3y^2} = 1 \], this translates to \[ 75 - 3y^2 = \ln(1) = 0 \] in logarithmic form because the natural logarithm of 1 is zero.

• Logarithms help in converting multiplicative relationships into additive ones, making them easier to handle mathematically.
• They are crucial in various scientific disciplines including sound measurement (decibels), earthquake intensity (Richter scale), and more.

Thus, using logarithms, we can simplify the problem significantly and solve for the variable \(y\).

Real solutions refer to answers or roots of equations that are real numbers. They differ from complex solutions, which include imaginary numbers. In real numbers, solutions are either rational (expressible as a fraction) or irrational (like square roots that can't be simplified further, such as \(\sqrt{2}\)).

• The solutions to the problem are \((x, y) = (0, 5)\) and \((x, y) = (0, -5)\), meaning both \(x\) and \(y\) are real numbers.
• Real solutions are crucial in many practical applications, where we need real-world numbers, like distance or time.

In solving our system of equations, we use these real solutions to understand the interaction between the different mathematical components given by the equations. Systems often test if there are real overlaps between values that satisfy all involved equations. If no real solutions exist, it indicates that the situation described cannot happen with real-world numbers.