Problem 33 - Algebra 2

Question

Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. $$ 4 x^{2}-8 x=5 $$

Step-by-Step Solution

Answer

The roots are located between the integers 0 and 1, and 2 and 3.

Step 1: Rewrite the Equation in Standard Form

The first step is to make sure the quadratic equation is in standard form, which is \( ax^2 + bx + c = 0 \). Starting with the equation \( 4x^2 - 8x = 5 \), subtract 5 from both sides to get \( 4x^2 - 8x - 5 = 0 \).

Step 2: Identify the Components of the Quadratic Equation

Identify the coefficients and constants from the quadratic equation. For \( 4x^2 - 8x - 5 = 0 \), we have \( a = 4 \), \( b = -8 \), and \( c = -5 \).

Step 3: Graph the Quadratic Function

Graph the quadratic function \( y = 4x^2 - 8x - 5 \) using graphing software or graphing calculator. The roots of the equation will be the x-values where the graph intersects the x-axis.

Step 4: Analyze the Graph for Roots

Look at the graph to find the x-intercepts. These are the points where the quadratic graph crosses the x-axis. The x-intercepts represent the roots of the equation. If they are not exact values, observe the closest integer values that contain the roots.

Step 5: State the Roots or Identify the Consecutive Integers

If you can determine the exact roots from the graph (where the curve crosses the x-axis), state those. Otherwise, identify the consecutive integers between which each root lies. For the graph of \( y = 4x^2 - 8x - 5 \), it appears the roots are between the integers 0 and 1, and 2 and 3 based on where the parabola seems to cross the x-axis.

Key Concepts

Graphing Quadratic Functions

Visualizing solutions by graphing is a helpful technique when dealing with quadratic equations. Here, we focus on the steps to graph a quadratic function and find its roots.

• First, ensure your quadratic equation is in the form of \( ax^2 + bx + c = 0 \). A graph of such an equation will form a curve called a parabola.
• Using either graphing software or a calculator, plot the equation. In our example, the function is \( y = 4x^2 - 8x - 5 \).
• When graphing, pay attention to the curve's shape and position. The parabola could open up or down depending on the sign of \( a \). In this case, since \( a = 4 \), it opens upwards.
• The x-intercepts of the graph, where the curve crosses the x-axis, signify the roots or solutions of the quadratic equation. Look closely at where these intersections happen, as they help determine the roots.

Understanding how to manually sketch or use a tool to graph can significantly aid in verifying your computational solutions.

Quadratic Equation Standard Form

A fundamental step in working with quadratic equations is placing them in their standard form. The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \). Here's how to ensure your equation is ready to be solved:

• Take your initial equation and rearrange it into the general format \( ax^2 + bx + c = 0 \). For example, start with \( 4x^2 - 8x = 5 \).


• Subtract 5 from both sides to acquire \( 4x^2 - 8x - 5 = 0 \).
• Next, identify the values of \( a \), \( b \), and \( c \). For \( 4x^2 - 8x - 5 = 0 \), these are \( a = 4 \), \( b = -8 \), and \( c = -5 \).

Once you have your equation in this format, it becomes much easier to apply different solving techniques, such as factoring, using the quadratic formula, or graphing. The clearer your standard form is, the simpler the solving process becomes.

Roots of a Quadratic Equation

Finding the roots of a quadratic equation reveals the solutions where the function equals zero. Graphing is a practical approach to identify these roots, but understanding their basic properties is crucial too.

• A quadratic function can have either two roots, one root, or no real roots. These roots correspond to the points where the graph intersects the x-axis, known as x-intercepts.
• The nature of these roots can sometimes indicate if they are real or complex. If the parabola crosses the x-axis, the roots are real and distinct. If it only touches the axis, there is a double root.


• If graphing, determine if the exact integer values can be observed or if they're approximate. With our example, the parabola intersects between the integers 0 and 1, and also between 2 and 3. These intervals suggest that the roots are located somewhere within these ranges.

Grasping the concept of roots in quadratic equations lays the groundwork for deeper algebraic problem-solving skills, as these points often feature prominently in mathematical applications.