Problem 56 - Algebra 2

Question

Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. \(0=-x^{2}-4 x+5\)

Step-by-Step Solution

Answer

Roots are between -5 and -4, and between 0 and 1.

Step 1: Rewrite the Equation into Standard Form

Start with the given equation: \[ 0 = -x^2 - 4x + 5 \] To make it easier to graph, rewrite this in standard quadratic form, which is \[ y = ax^2 + bx + c \] Here, we have \[ y = -x^2 - 4x + 5 \] Now it's ready to graph.

Step 2: Identify Quadratic Properties

Identify the components of the quadratic equation that will help with graphing:- The quadratic coefficient (\( a \)) is \(-1\), indicating the parabola opens downwards.- The vertex form is not directly given, but from \( ax^2 + bx + c \), it's clear the equation is already set.- The \( y \)-intercept is at \( (0, 5) \) since when \( x = 0 \), \( y = 5 \).

Step 3: Graph the Parabola

Plot the graph of the equation \( y = -x^2 - 4x + 5 \) using its properties:- Start from the \( y \)-intercept (0, 5).- Since the parabola opens downwards, and knowing it’s symmetric about the vertex line, plot additional points. - Alternatively, use graphing software or graph paper to ensure a smooth curve.

Step 4: Identify Roots of the Equation

The roots of the equation are the \( x \)-values where \( y = 0 \) (i.e., where the graph intersects the \( x \)-axis):- By visual inspection of the graph, identify that these points lie between \(-5 \leq x \leq -3 \) and \( 1 \leq x \leq 3 \).

Step 5: Confirm the Consecutive Interval

Estimate the locations further by observing where the parabola dips below the \( x \)-axis:- From inspection, confirm the roots lie between \(-5 \) and \(-4 \), as well as between \( 0 \) and \( 1 \). These are the intervals where the expression likely goes from positive to negative and vice versa.

Key Concepts

Graphing Quadratics

Graphing a quadratic equation is a visual way of finding its roots. The standard form of a quadratic equation is given as \( y = ax^2 + bx + c \). This specific equation, \( y = -x^2 - 4x + 5 \), involves a few critical steps for graphing.

• Identify the quadratic term (\( ax^2 \)): Here, \( a = -1 \), which tells us that the parabola opens downwards.
• Plot the y-intercept, where \( x = 0 \): In our example, it's at (0, 5).
• Draw the parabola, using symmetry around its vertex to ensure the graph is accurate.
• For more precise parabolas, equidistant points from the vertex can be calculated and plotted.

Graphing helps in visualizing where the quadratic may cross the x-axis, indicating possible roots.

Quadratic Roots

Quadratic roots are the values of \( x \) where the graph crosses the x-axis. These are the solutions to the equation \( ax^2 + bx + c = 0 \).
In our equation, the roots are where \( y = 0 \), so we find when \( -x^2 - 4x + 5 = 0 \). These intersections are critical:

• For exact values, observe the x-coordinates at which the parabola intersects the x-axis.
• If graphing visually (without calculations), estimate the range by identifying the closes points where the parabola approaches the x-axis.
• Using a software tool or closely analyzing the graph can provide a more accurate estimation of the roots.
• Our graph shows these points fall approximately between \(-5\) and \(-4\), and between \(0\) and \(1\).

Identifying roots is essential, as they represent the solutions to the quadratic equation.

Parabola Properties

The properties of a parabola are derived from its quadratic equation and help understand its shape and behavior.
The equation \( y = -x^2 - 4x + 5 \) shows key traits through its components:

• Direction: The parabola opens downwards since the coefficient of \( x^2 \) is negative.
• Vertex: Though not calculated here, the vertex can be found using the formula \( x = -\frac{b}{2a} \).
• Y-Intercept: This is the point where the parabola crosses the y-axis, at (0, 5).
• Symmetry: The parabola is symmetric about the vertical line passing through the vertex.

Understanding these properties allows one to accurately sketch the parabola, capture its essence, and deduce additional features like the maximum or minimum points. Knowing how a quadratic functions behave aids in solving many real-world mathematical problems.