Problem 13
Question
Solve each equation. For equations with real solutions, support your answers graphically. $$x^{2}=16$$
Step-by-Step Solution
Verified Answer
The solutions are \( x = 4 \) and \( x = -4 \).
1Step 1: Identify the Type of Equation
The given equation is \( x^2 = 16 \), which is a basic quadratic equation of the form \( x^2 = c \) where \( c \) is a constant.
2Step 2: Solve the Equation Algebraically
To solve the quadratic equation \( x^2 = 16 \), take the square root of both sides. This gives us two solutions: \( x = \sqrt{16} \) and \( x = -\sqrt{16} \), which simplifies to \( x = 4 \) and \( x = -4 \).
3Step 3: Verify the Solutions Graphically
Consider the function \( f(x) = x^2 \) and its graph. The solutions to the equation \( x^2 = 16 \) are the \( x \)-values where the graph of \( y = x^2 \) intersects the horizontal line \( y = 16 \). The graph confirms the intersections at \( x = 4 \) and \( x = -4 \).
4Step 4: Write Down the Solutions
Based on our algebraic and graphical analysis, the solutions to the equation \( x^2 = 16 \) are \( x = 4 \) and \( x = -4 \).
Key Concepts
Square RootGraphical SolutionsAlgebraic Solutions
Square Root
The square root is a fundamental concept in solving quadratic equations such as \( x^2 = 16 \). A square root of a number \( c \) is a value that, when multiplied by itself, gives \( c \). For the equation \( x^2 = 16 \), we find the square root of both sides to solve the equation.
This operation generates two possible solutions because both positive and negative numbers squared result in a positive number. Therefore, both \( x = \sqrt{16} \) and \( x = -\sqrt{16} \) are valid, leading to solutions \( x = 4 \) and \( x = -4 \). This dual nature is crucial to remember when taking square roots in quadratic equations, as neglecting the negative square root could lead to missing a solution.
In more complex equations, solving to find the square root might involve decimals or rational numbers, but the fundamental principle remains the same.
This operation generates two possible solutions because both positive and negative numbers squared result in a positive number. Therefore, both \( x = \sqrt{16} \) and \( x = -\sqrt{16} \) are valid, leading to solutions \( x = 4 \) and \( x = -4 \). This dual nature is crucial to remember when taking square roots in quadratic equations, as neglecting the negative square root could lead to missing a solution.
In more complex equations, solving to find the square root might involve decimals or rational numbers, but the fundamental principle remains the same.
Graphical Solutions
Graphical solutions are a powerful technique to visualize and confirm the solutions of quadratic equations. When working with equations like \( x^2 = 16 \), imagining or drawing the function \( y = x^2 \) can assist a lot.
Here’s how it works: You plot the function \( y = x^2 \) on a coordinate plane. This forms a characteristic parabolic curve opening upwards. Then, plot the horizontal line \( y = 16 \). The points where this parabola intersects the horizontal line are your solutions – which graphically confirms our solutions \( x = 4 \) and \( x = -4 \) from the earlier algebraic method.
This method can be very helpful in understanding the nature of quadratic equations intuitively, as it visually illustrates that the equation has two real solutions. Graphing tools and software can make this process easier and enhance comprehension.
Here’s how it works: You plot the function \( y = x^2 \) on a coordinate plane. This forms a characteristic parabolic curve opening upwards. Then, plot the horizontal line \( y = 16 \). The points where this parabola intersects the horizontal line are your solutions – which graphically confirms our solutions \( x = 4 \) and \( x = -4 \) from the earlier algebraic method.
This method can be very helpful in understanding the nature of quadratic equations intuitively, as it visually illustrates that the equation has two real solutions. Graphing tools and software can make this process easier and enhance comprehension.
Algebraic Solutions
Solving quadratic equations algebraically involves systematically manipulating the equation to find the unknown variable. Using the equation \( x^2 = 16 \), solve it by isolating the variable \( x \).
The primary step is taking the square root of both sides of the equation in order to find \( x \). By doing so, the equation simplifies as follows:
While this method relies heavily on following algebraic rules, it's one of the core ways to solve equations efficiently, particularly when equations grow more complex. Algebraic solutions grant precision and are crucial for equations where graphical or numerical methods may be cumbersome.
The primary step is taking the square root of both sides of the equation in order to find \( x \). By doing so, the equation simplifies as follows:
- Taking \( \sqrt{x^2} = \sqrt{16} \)
- This leads to \( x = 4 \) and \( x = -4 \)
While this method relies heavily on following algebraic rules, it's one of the core ways to solve equations efficiently, particularly when equations grow more complex. Algebraic solutions grant precision and are crucial for equations where graphical or numerical methods may be cumbersome.
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