Solve the quadratic equation graphically using at least two different approaches. When necessary, give your solutions to the nearest hundredth.
[tex]2 x^2-3=2 x[/tex]
a. [tex]x=-1.1[/tex] or [tex]x=-1[/tex]
b. [tex]x=1.82[/tex] or [tex]x=-0.82[/tex]
c. [tex]x=0.10[/tex] or [tex]x=-3[/tex]
d. [tex]x=0[/tex] or [tex]x=-6.3[/tex]
Please select the best answer from the choices provided
Answer (2)
The solutions to the quadratic equation 2 x 2 − 2 x − 3 = 0 are approximately x = 1.82 and x = − 0.82 . Using two different approaches, we graphed the quadratic and found the x-intercepts as well as intersecting a linear function. The correct answer is Option B.
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Rewrite the equation as 2 x 2 − 2 x − 3 = 0 .
Graph y = 2 x 2 − 2 x − 3 and find the x-intercepts, or graph y = 2 x 2 − 3 and y = 2 x and find the intersection points.
Approximate the solutions to the nearest hundredth.
The solutions are x = 1.82 or x = − 0.82 , so the answer is B .
Explanation
Understanding the Problem We are given the quadratic equation 2 x 2 − 3 = 2 x . Our goal is to solve it graphically using two different approaches and choose the correct answer from the provided options.
Rewriting the Equation First, let's rewrite the equation in the standard quadratic form: 2 x 2 − 2 x − 3 = 0 .
Approach 1: Finding X-Intercepts Approach 1: Graphing y = 2 x 2 − 2 x − 3 and finding the x-intercepts. The x-intercepts are the points where the graph crosses the x-axis (i.e., where y = 0 ). These points represent the solutions to the equation 2 x 2 − 2 x − 3 = 0 .
Approach 2: Finding Intersection Points Approach 2: Graphing y = 2 x 2 − 3 and y = 2 x on the same coordinate plane and finding their intersection points. The x-coordinates of the intersection points are the solutions to the equation 2 x 2 − 3 = 2 x .
Finding the Roots Using a calculator or a graphing tool, we can find the approximate roots of the equation 2 x 2 − 2 x − 3 = 0 . The roots are approximately x = − 0.82 and x = 1.82 .
Selecting the Correct Answer Comparing the solutions we found graphically with the given options, we see that option B, x = 1.82 or x = − 0.82 , matches our solutions.
Final Answer Therefore, the best answer is B.
Examples
Quadratic equations are used in various real-world applications, such as determining the trajectory of a projectile, calculating the dimensions of a rectangular area given its perimeter and area, and modeling the growth or decay of populations. For example, if you throw a ball, the height of the ball over time can be modeled by a quadratic equation. Solving the equation helps determine when the ball will hit the ground.
