Problem 37 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The roots are between 0 and 1, and between 3 and 4.

Step 1: Understand the Equation

The equation given is \(x^{2} - 4x + 2 = 0\), which is a quadratic equation in the standard form \(ax^{2} + bx + c = 0\). Here, \(a = 1\), \(b = -4\), and \(c = 2\).

Step 2: Graph the Quadratic Function

Construct the graph of the function \(y = x^{2} - 4x + 2\). This is a parabola that opens upwards because the coefficient of \(x^{2}\) is positive.

Step 3: Plot Key Points on the Graph

To plot the graph, find the vertex of the parabola using the formula \(x = -\frac{b}{2a}\), which gives \(x = 2\). Calculate the y-coordinate of the vertex as \(y = (2)^2 - 4(2) + 2= -2\). The vertex of the parabola is (2, -2).

Step 4: Determine the Y-Intercept

The y-intercept can be found by setting \(x = 0\) in the equation \(y = x^{2} - 4x + 2\). Thus, \(y = 0^{2} - 4(0) + 2 = 2\). Plot this point (0, 2) on the graph.

Step 5: Check Additional Points for Curvature

Pick additional points to accurately graph the parabola, such as \(x = 1\) and \(x = 3\). For \(x = 1\), \(y = 1^2 - 4(1) + 2 = -1\); for \(x = 3\), \(y = 3^2 - 4(3) + 2 = -1\). Plot these points (1, -1) and (3, -1).

Step 6: Find the Roots by Identifying the X-Intercepts

Observe where the plotted graph of the parabola intersects the x-axis. These points are the roots or solutions to the equation \(x^{2} - 4x + 2 = 0\).

Step 7: Analyze the Graph for Exact Roots

If the graph shows that the parabola intersects the x-axis at non-integer locations, estimate the x-values of these intersections. For this equation, the parabola crosses between the integers 0 and 1, and also between 3 and 4.

Step 8: Conclude the Root Location

Since the graph does not provide exact integer roots, the roots of the equation are approximately between 0 and 1, and also between 3 and 4.

Key Concepts

Graphing Parabolas

Graphing a parabola is a crucial technique when working with quadratic equations. A quadratic equation typically takes the form \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. The graph of this equation will be a U-shaped curve called a parabola. Parabolas can open upwards or downwards depending on the sign of \(a\).

• If \(a > 0\), the parabola opens upwards, like a smile.
• If \(a < 0\), it opens downwards, like a frown.

When graphing, identifying key points like the vertex, y-intercept, and additional points on either side of the vertex is essential for accuracy. Begin by plotting the vertex, then the y-intercept, and at least two additional points to reveal the curve's shape and direction.

Finding X-Intercepts

The x-intercepts of a parabola are the points where the graph crosses the x-axis. These points are also known as the roots or solutions of the quadratic equation \(ax^2 + bx + c = 0\).
To find the x-intercepts graphically:

• First, graph the quadratic function.
• Look for the points where the parabola crosses the x-axis.

These intersections reflect the roots of the equation. If the parabola touches the x-axis at a single point, it has one root (a double root). If it doesn't intersect, there are no real roots. In the given problem, the parabola intersects between 0 and 1, and again between 3 and 4, suggesting root intervals.

Vertex of a Parabola

The vertex of a parabola is a critical point that helps define its shape. It is the point where the parabola changes direction, either from upwards to downwards or downwards to upwards.
To find the vertex, you use the formula \(x = -\frac{b}{2a}\), where \(a\) and \(b\) are coefficients from the equation \(ax^2 + bx + c\). Once you have \(x\), substitute it back into the equation to find \(y\).
In the given problem, the calculated vertex is at \((2, -2)\). The vertex is a minimum point for parabolas that open upwards and a maximum point for parabolas that open downwards. This problem's parabola opens upwards, making \((2, -2)\) its lowest point.

Y-Intercept

The y-intercept of a quadratic function is another key component when graphing a parabola. It is the point where the graph intersects the y-axis. This occurs when \(x = 0\).
To find the y-intercept, simply substitute \(x = 0\) into the equation \(y = ax^2 + bx + c\). Calculating the value gives you the coordinate (0, \(c\)).
For the equation \(x^2 - 4x + 2 = 0\), you set \(x = 0\), resulting in \(y = 2\). Therefore, the y-intercept is at the point (0, 2). This point provides another known location on the graph, which helps shape the parabola alongside the vertex and additional plotted points.