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Magazine Chapter 8: Problem 2
$$ y=x^{2}-2 \text { and } y=0 $$
Short Answer
Expert verified
The solutions are \( x = \sqrt{2} \) and \( x = -\sqrt{2} \).
Step by step solution
01
Understand the Problem
The exercise consists of two equations: 1. \( y = x^2 - 2 \) 2. \( y = 0 \)We need to find the values of \( x \) where both equations are satisfied simultaneously. Essentially, this is finding the points where the quadratic function intersects the x-axis.
02
Set the Equations Equal
To find the solution, set the first equation equal to the second: \[ x^2 - 2 = 0 \] This will allow us to solve for the values of \( x \) that satisfy both equations.
03
Solve for x
Solve the equation \( x^2 - 2 = 0 \) for \( x \). First, add 2 to both sides of the equation to isolate \( x^2 \): \[ x^2 = 2 \]Next, take the square root of both sides to solve for \( x \):\[ x = \pm \sqrt{2} \] This gives us the solutions \( x = \sqrt{2} \) and \( x = -\sqrt{2} \).
04
Verify the Solution
Substitute \( x = \sqrt{2} \) and \( x = -\sqrt{2} \) back into the original equation \( y = x^2 - 2 \) to verify the solutions:For \( x = \sqrt{2} \): \[ y = (\sqrt{2})^2 - 2 = 2 - 2 = 0 \]For \( x = -\sqrt{2} \): \[ y = (-\sqrt{2})^2 - 2 = 2 - 2 = 0 \]Both values satisfy \( y = 0 \), confirming the solutions are correct.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Parabolas
A parabola is a type of curve on a graph. It is the graph that represents a quadratic equation, which is generally in the form of \( y = ax^2 + bx + c \). Parabolas have distinct characteristics:
- Vertex: The highest or lowest point of the parabola, depending on whether it opens upwards or downwards. In our example \( y = x^2 - 2 \), the vertex is at the point (0,-2).
- Axis of Symmetry: A line that runs through the vertex, dividing the parabola into two identical halves. For the equation \( y = x^2 - 2 \), the axis of symmetry is the y-axis or \( x = 0 \).
- Opening Direction: A parabola can open upwards (positive \( a \)) or downwards (negative \( a \)). Here, the coefficient of \( x^2 \) is positive, so it opens upwards.
Roots of Equations
The roots of a quadratic equation are the values of \( x \) for which \( y = 0 \). They are also known as "solutions" or "zeros" of the equation. Finding the roots essentially means determining where the graph of the equation touches or crosses the x-axis.
In our problem, we found the roots by solving \( x^2 - 2 = 0 \). By solving, we determined that the roots are \( x = \sqrt{2} \) and \( x = -\sqrt{2} \).
Roots can sometimes be rational, like \( x = 2 \) or \( x = -1 \), or irrational, as in our example, \( \sqrt{2} \).
In our problem, we found the roots by solving \( x^2 - 2 = 0 \). By solving, we determined that the roots are \( x = \sqrt{2} \) and \( x = -\sqrt{2} \).
Roots can sometimes be rational, like \( x = 2 \) or \( x = -1 \), or irrational, as in our example, \( \sqrt{2} \).
- Real Roots: When the graph physically touches or crosses at these points on the x-axis.
- Imaginary Roots: When the graph doesn't intersect with the x-axis, which happens with negative discriminants.
Solving Equations
Solving a quadratic equation involves finding the values of \( x \) that make the equation true. There are several methods to solve such equations:
Choosing the method depends on the original form of the equation and the ease with which the solution can be extracted. Practice with different approaches will improve your problem-solving skills.
- Factoring: Expressing the quadratic in its factorized form and setting each factor to zero. However, our equation does not factor neatly.
- Completing the Square: A method that transforms the equation into a perfect square trinomial.
- Quadratic Formula: A universal method employing the formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), although in our example, the simpler method of solving \( x^2 - 2 = 0 \) directly was optimal.
Choosing the method depends on the original form of the equation and the ease with which the solution can be extracted. Practice with different approaches will improve your problem-solving skills.
