Problem 21 - Algebra 1

Question

Find the \(x\) -intercepts of the graph of the equation. $$y=4 x^{2}+8 x-1$$

Step-by-Step Solution

Answer

The x-intercepts of the function \(y = 4x^2 + 8x - 1\) are \(x = -1 + \sqrt{5}\) and \(x = -1 - \sqrt{5}\)

Step 1: Set the equation to zero

To find the x-intercepts, first set the equation to zero: \(0 = 4x^2 + 8x - 1\)

Step 2: Re-arrange the equation into standard form

Re-arrange the equation to a format compatible with the quadratic formula, \(ax^2 + bx + c = 0\), which in this case it is already in the standard form: \(0 = 4x^2 + 8x - 1\) where \(a = 4\), \(b = 8\), and \(c = -1\)

Step 3: Use the quadratic formula

Use the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to solve for \(x\). This results in \(x = \frac{-8 \pm \sqrt{(8)^{2} - 4*4*(-1)}}{2*4}\)

Step 4: Simplify the solution

Simplify within the square root first: \(x = \frac{-8 \pm \sqrt{64 - (-16)}}{8}\). Then simplify further: \(x = \frac{-8 \pm \sqrt{80}}{8}\). This simplifies to \(x = -1 \pm \sqrt{5}\)

Key Concepts

Understanding x-intercepts

The x-intercepts of a graph are the points where the graph crosses the x-axis. At these points, the value of y is zero. Solving for the x-intercepts means finding the x-values where this occurs.
When given an equation like \(y = 4x^2 + 8x - 1\), to find the x-intercepts, set y equal to zero and solve for x:

• Start by setting the equation to zero: \(0 = 4x^2 + 8x - 1\)
• At this stage, you transform the problem into solving a quadratic equation, where the y-value is zero
• This step is critical because only then can you find the points along the x-axis where the curve intersects

X-intercepts are significant because they provide important information about the roots of the equation. Understanding x-intercepts can also help in sketching the graph of the quadratic function.

Using the quadratic formula

The quadratic formula is a powerful tool for solving quadratic equations in the form \(ax^2 + bx + c = 0\). The formula allows us to find the x-values, or roots, where the curve touches or crosses the x-axis:

• The formula itself is: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
• Using this formula involves identifying the coefficients \(a\), \(b\), and \(c\) from the quadratic equation
• This process requires a basic understanding of each component's influence in the equation solution

In our specific problem, the equation \(0 = 4x^2 + 8x - 1\) gives us \(a = 4\), \(b = 8\), and \(c = -1\). Applying these values to the quadratic formula helps identify the x-intercepts:

• Compute \(b^2 - 4ac\) as the discriminant, then proceed with solving under the square root
• Simplify step by step until you reach the neatest form: \(x = -1 \pm \sqrt{5}\)

The quadratic formula is especially helpful because it offers a straightforward pathway to the solution, even when factoring is complex or impossible.

Rearranging into standard form

The standard form of a quadratic equation is \(ax^2 + bx + c = 0\). This form is key for applying the quadratic formula correctly. To identify or confirm that an equation like \(y = 4x^2 + 8x - 1\) is in standard form:

• Check if it's arranged as \(ax^2 + bx + c = 0\)
• Ensure the equation equals zero, for example, rearrange \(y = 4x^2 + 8x - 1\) as \(0 = 4x^2 + 8x - 1\)
• Determine and assign values to \(a\), \(b\), and \(c\)

Recognizing the standard form is crucial for applying the quadratic formula, as it simplifies the process of substituting the values into the formula directly.
Utilizing this step ensures accuracy and ease when determining the x-intercepts of the quadratic function.