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Chapter 6: Problem 94

Use a graphing calculator to find any solutions that exist accurate to two decimal places. $$ x^{2}+4.68=1.2 x $$

Short Answer

Expert verified
The solutions are complex since the graph does not intersect the x-axis.

Step by step solution

01

Rewrite the Equation

First, rewrite the equation in standard form: \[ x^2 - 1.2x + 4.68 = 0 \]
02

Enter the Equation into the Graphing Calculator

Enter the equation \( y = x^2 - 1.2x + 4.68 \) into the graphing calculator to find the points where the graph intersects the x-axis. These points are your solutions.
03

Find the Intercepts

Use the graphing calculator to find the x-intercepts (roots) of the equation. Adjust the graph's zoom if necessary to identify the points accurately.
04

Note the Solutions

Record the x-values of the points where the graph intersects the x-axis. Ensure that the solutions are accurate to two decimal places.

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Key Concepts

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

standard form of a quadratic equation
To solve a quadratic equation using a graphing calculator, you first need to rewrite the equation in its standard form. The standard form of a quadratic equation is:
ewline ewline ax^{2} + bx + c = 0.
ewline In the given exercise, the equation is initially:
x^{2} + 4.68 = 1.2x.
ewline Reorganize it by moving all terms to one side to form:
x^{2} - 1.2x + 4.68 = 0.
ewline Now it's in standard form, where:
ewline
  • a = 1
  • b = -1.2
  • c = 4.68
ewline This is the essential starting point for using a graphing calculator.
x-intercepts
Once the quadratic equation is in standard form, it's time to find the solutions, or x-intercepts.
ewline x-intercepts are the values of x where the graph of the equation crosses the x-axis. This happens where the equation equals zero.
ewline For the quadratic equation
x^{2} - 1.2x + 4.68 = 0,
ewline we are looking for the points on the graph where y = 0.
ewline In mathematical terms, finding the x-intercepts means solving for the values of x that satisfy the equation. Using a graphing calculator, these points can easily be visualized.
graphing calculator usage
A graphing calculator is a very useful tool for solving quadratic equations, especially for finding x-intercepts.
ewline Here鈥檚 how to use it:
ewline
  • Turn on the graphing calculator.
  • Enter the quadratic equation in the form `y = x^{2} - 1.2x + 4.68`.
  • Use the graph function to plot the equation.
  • Look at the graph to see where it crosses the x-axis.
  • Use the calculator鈥檚 built-in functions to find the exact x-intercepts, usually found under a 'Calc' or 'Zero' menu.
  • Adjust zoom settings if necessary to clearly see the intercepts.
ewline Graphing calculators make it much easier to visualize and solve quadratic equations.
quadratic functions
Quadratic functions are a type of polynomial function that take the form: ewline ewline f(x) = ax^{2} + bx + c
ewline Here, the highest degree of x is 2. These functions create a parabola when graphed.
ewline Key characteristics of quadratic functions include:
ewline
  • The graph is a U-shaped curve called a parabola.
  • The direction of the parabola (upward or downward) is determined by the coefficient a. If a > 0, it opens upwards; if a < 0, it opens downwards.
  • The vertex is the highest or lowest point on the graph, depending on its direction.
  • The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetric parts.
  • x-intercepts (roots) are the points where the graph crosses the x-axis.
ewline Understanding these properties helps in graphing and solving quadratic equations using a graphing calculator.

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Most popular questions from this chapter

Write an equivalent expression by factoring out the smallest power of \(x\) in each of the following. $$ x^{3 / 4}+x^{1 / 2}-x^{1 / 4} $$

Find the zeros of each function. $$ f(x)=2 x^{2}+4 x+2 $$

What is wrong with solving \(x^{2}=3 x\) by dividing both sides of the equation by \(x ?\)

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ a^{2}-8 a+16-b^{2} $$

Factor completely. Assume that variables in exponents represent positive integers. $$ 0.09 x^{8}+0.48 x^{4}+0.64 $$
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