Problem 46
Question
REASONING Examine the graph of \(y=x^{2}-4 x-5\) a. What are the solutions of \(0=x^{2}-4 x-5 ?\) b. What are the solutions of \(x^{2}-4 x-5 \geq 0 ?\) c. What are the solutions of \(x^{2}-4 x-5 \leq 0 ?\)
Step-by-Step Solution
Verified Answer
a) \(x = 5\) and \(x = -1\); b) \(( -\infty, -1 ] \cup [ 5, \infty )\); c) \([-1, 5]\)
1Step 1: Identify the Equation
We are dealing with the quadratic equation \(y = x^2 - 4x - 5\). We need to find the solutions for various conditions on this equation.
2Step 2: Solve for the Roots (a)
To find the solutions of \(0 = x^2 - 4x - 5\), set the quadratic equation equal to zero, resulting in \(x^2 - 4x - 5 = 0\). Factor the equation as \((x - 5)(x + 1) = 0\). Therefore, the roots are \(x = 5\) and \(x = -1\).
3Step 3: Analyze Inequality for (b)
For the inequality \(x^2 - 4x - 5 \geq 0\), examine the intervals where this holds true. The quadratic is greater than or equal to zero on the intervals where the product of its factors is greater than or equal to zero. The solution is \(( -\infty, -1 ] \cup [ 5, \infty )\).
4Step 4: Analyze Inequality for (c)
For the inequality \(x^2 - 4x - 5 \leq 0\), consider where the expression is less than or equal to zero. This occurs between the roots in the interval where the product of its factors is non-positive. The solution is \([-1, 5]\).
Key Concepts
Roots of EquationsInequalitiesFactorization
Roots of Equations
Understanding the roots of an equation is fundamental when dealing with quadratic equations like \( y = x^2 - 4x - 5 \). Roots are the values of \( x \) for which the equation equals zero. In the equation \( x^2 - 4x - 5 = 0 \), finding the roots means determining the points where this expression intersects the x-axis in its graph.
To find these roots, we often use factorization. This method simplifies the quadratic equation into a product of two linear factors. For example, we can express \( x^2 - 4x - 5 \) as \( (x - 5)(x + 1) = 0 \).
Each factor can be set separately to zero, giving us the roots. Hence, from \( (x - 5) = 0 \), we get \( x = 5 \), and from \( (x + 1) = 0 \), we get \( x = -1 \). Therefore, the roots of this equation are \( x = 5 \) and \( x = -1 \). These points can be plotted on the graph of the equation as the points where the curve crosses the x-axis.
To find these roots, we often use factorization. This method simplifies the quadratic equation into a product of two linear factors. For example, we can express \( x^2 - 4x - 5 \) as \( (x - 5)(x + 1) = 0 \).
Each factor can be set separately to zero, giving us the roots. Hence, from \( (x - 5) = 0 \), we get \( x = 5 \), and from \( (x + 1) = 0 \), we get \( x = -1 \). Therefore, the roots of this equation are \( x = 5 \) and \( x = -1 \). These points can be plotted on the graph of the equation as the points where the curve crosses the x-axis.
Inequalities
Quadratic inequalities involve expressions like \( x^2 - 4x - 5 \geq 0 \) and \( x^2 - 4x - 5 \leq 0 \). Solving these involves finding intervals on the x-axis where the inequality holds true.
To solve \( x^2 - 4x - 5 \geq 0 \), first identify the roots, \( x = -1 \) and \( x = 5 \), which act as boundary points. Analyze intervals created by these roots: \(( -\infty, -1 )\), \([-1, 5]\), and \((5, \infty)\).
In \(( -\infty, -1 ]\) and \([ 5, \infty )\), the quadratic expression is either zero or positive. This is where the graph of the quadratic is on or above the x-axis, so the solution is \(( -\infty, -1 ] \cup [ 5, \infty )\).
For the inequality \( x^2 - 4x - 5 \leq 0 \), observe that it holds between the roots, in \([-1, 5]\). Here, the quadratic expression is zero or negative, represented by the section of the graph below or on the x-axis. This defines the interval \([-1, 5]\) as the solution.
To solve \( x^2 - 4x - 5 \geq 0 \), first identify the roots, \( x = -1 \) and \( x = 5 \), which act as boundary points. Analyze intervals created by these roots: \(( -\infty, -1 )\), \([-1, 5]\), and \((5, \infty)\).
In \(( -\infty, -1 ]\) and \([ 5, \infty )\), the quadratic expression is either zero or positive. This is where the graph of the quadratic is on or above the x-axis, so the solution is \(( -\infty, -1 ] \cup [ 5, \infty )\).
For the inequality \( x^2 - 4x - 5 \leq 0 \), observe that it holds between the roots, in \([-1, 5]\). Here, the quadratic expression is zero or negative, represented by the section of the graph below or on the x-axis. This defines the interval \([-1, 5]\) as the solution.
Factorization
Factorization is a crucial concept in solving quadratic equations, particularly when identifying roots. It simplifies the equation by splitting it into two binomial expressions. For a general quadratic \( ax^2 + bx + c = 0 \), factorization involves expressing it as \((mx + n)(px + q) = 0\), where the product equal to zero implies either \(mx + n = 0\) or \(px + q = 0\).
For the quadratic \( x^2 - 4x - 5 \), factors are found as \( (x - 5)(x + 1)\). This shows that once expanded, the terms satisfy \( a = 1, b = -4, \) and \( c = -5 \).
Each individual factor gives us a root. Solving \( x - 5 = 0 \) results in \( x = 5 \), and \( x + 1 = 0 \) gives \( x = -1 \).
Factorization not only helps in determining the roots but is a foundational skill for solving more complex inequalities and verifying solutions in mathematics. It is particularly handy for quadratic equations that can be easily broken into simpler expressions.
For the quadratic \( x^2 - 4x - 5 \), factors are found as \( (x - 5)(x + 1)\). This shows that once expanded, the terms satisfy \( a = 1, b = -4, \) and \( c = -5 \).
Each individual factor gives us a root. Solving \( x - 5 = 0 \) results in \( x = 5 \), and \( x + 1 = 0 \) gives \( x = -1 \).
Factorization not only helps in determining the roots but is a foundational skill for solving more complex inequalities and verifying solutions in mathematics. It is particularly handy for quadratic equations that can be easily broken into simpler expressions.
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