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Chapter 9: Problem 54

Find the x-intercepts of the graph of the equation. $$y=2 x^{2}-6 x-8$$

Short Answer

Expert verified
The x-intercepts of the graph of the equation are \(x = 4\) and \(x = -1\)

Step by step solution

01

Write Down the Quadratic Equation

The given quadratic equation is \(y = 2x^{2} - 6x - 8\).
02

Set y to 0 and Make the Equation Standard

We are looking for the points where the equation crosses the x-axis. For this, we set y = 0, because the y-value at the x-axis is 0. We can write down the following equation: \(0 = 2x^{2} - 6x - 8\)
03

Find the x-Intercepts by Solving Quadratic Equation

We solve the equation \(0 = 2x^{2} - 6x - 8\) for \(x\). First, divide the equation by two to simplify it, which gives \(0 = x^{2} - 3x - 4\). Then we factorize which gives \((x-4)(x+1) = 0\). Setting each factor to zero gives the solutions \(x = 4\) and \(x = -1\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

x-intercepts
When dealing with graphs of equations, x-intercepts are key points that highlight where the graph of a function touches or crosses the x-axis. In simple terms, x-intercepts are the coordinates on the graph where the value of y is zero. To find x-intercepts, follow these steps:
  • Set the entire equation equal to zero. For example, if you have an equation such as \(y = 2x^2 - 6x - 8\), to find its x-intercepts, we set \(y = 0\) resulting in \(0 = 2x^2 - 6x - 8\).
  • Solve the resulting equation in terms of \(x\). This involves techniques like factoring or using the quadratic formula.
Finding x-intercepts is essential to understand the behavior of quadratic equations, as they give points where the graph precisely hits the x-axis.
factoring quadratics
Factoring quadratics is a process used to rewrite a quadratic equation as a product of its linear factors. This is often a simpler form that makes solving the equation easier.To factor quadratic equations like \(0 = x^2 - 3x - 4\), you need to:
  • Identify two numbers that multiply together to give the constant term (in this case, \(-4\)) and add up to give the linear coefficient (\(-3\)). Here, those numbers are \(1\) and \(-4\).
  • Rewrite the quadratic as a product of binomials: \((x-4)(x+1) = 0 \).
This factorization process is crucial as it allows you to quickly find the solutions of the quadratic equation by setting each factor equal to zero.
solving quadratic equations
Solving quadratic equations involves finding values of \(x\) that satisfy the equation. This can be done through different methods, such as factoring, completing the square, or using the quadratic formula. For the equation \(0 = 2x^2 - 6x - 8\), we focused on:**Simplifying and Factoring**
  • First, we simplify the equation by dividing all terms by 2, resulting in \(x^2 - 3x - 4 = 0\).
  • Next, we factor the simplified equation resulting in \((x-4)(x+1) = 0\).
**Solving for x**
  • Once factored, set each bracket equal to zero: \(x - 4 = 0\) and \(x + 1 = 0\).
  • Solve for \(x\), giving solutions \(x = 4\) and \(x = -1\).
Ultimately, solving quadratic equations gives us possible x-values that make \(y = 0\), thereby determining the points of the graph (x-intercepts) that cross the x-axis.

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