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Magazine Chapter 13: Problem 42
Use a graphing calculator to solve each system. Then confirm your answer algebraically. \(y=2 x^{2}+4 x\) \(y=-x^{2}-1\)
Short Answer
Expert verified
The points of intersection are (-1/3, -10/9) and (-1, -2).
Step by step solution
01
- Graphing the equations
Use a graphing calculator to plot the equations. The first equation is y = 2x^2 + 4x and the second equation is y = -x^2 - 1. Observe the points where the graphs intersect.
02
- Identify intersection points
From the graph, determine the points of intersection. These points are the values of x and y where both equations are equal.
03
- Set the equations equal
To confirm graphically identified solutions algebraically, set the equations equal to each other:2x^2 + 4x = -x^2 - 1.
04
- Combine like terms
Move all terms to one side to obtain:2x^2 + 4x + x^2 + 1 = 0 Combine like terms to get: 3x^2 + 4x + 1 = 0.
05
- Solve the quadratic equation
Use the quadratic formula x = [-b 卤 鈭�(b^2 - 4ac)] / 2a with a = 3, b = 4, and c = 1 to solve:x = [-4 卤 鈭�(4^2 - 4(3)(1))] / 2(3).This simplifies to: x = [-4 卤 鈭�(16 - 12)] / 6 or x = [-4 卤 2] / 6, giving the solutionsx = -1/3 or x = -1.
06
- Find corresponding y-values
Substitute x = -1/3 and x = -1 back into one of the original equations to find the y-values. Using y = 2x^2 + 4x,For x = -1/3: y = 2(-1/3)^2 + 4(-1/3) = 2(1/9) - 4/3 = 2/9 - 12/9 = -10/9.For x = -1: y = 2(-1)^2 + 4(-1) = 2 - 4 = -2.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
graphing calculator
To solve systems of equations, a graphing calculator can be incredibly useful. This tool allows you to visually plot functions and identify their intersection points.
When using a graphing calculator, follow these steps:
When using a graphing calculator, follow these steps:
- Enter the given equations into the calculator.
- Make sure to set the viewing window appropriately to see where the plots intersect.
- Analyze the graph to find the intersection points. These are the x and y values that solve both equations simultaneously.
quadratic equations
Quadratic equations are polynomial equations of the second degree, generally represented as ax虏 + bx + c = 0.
These types of equations form parabolas when graphed.
In the current exercise, we have quadratic equations y = 2x虏 + 4x and y = -x虏 - 1.
These types of equations form parabolas when graphed.
In the current exercise, we have quadratic equations y = 2x虏 + 4x and y = -x虏 - 1.
- A quadratic equation typically has two solutions because of its parabolic shape.
- The solutions can be real numbers, imaginary, or complex depending on the discriminant \(\bbox[yellow]{b虏 - 4ac}\).
intersection points
Intersection points are the points where two or more graphs meet.
These points are important in solving a system of equations because they provide the solutions for both equations.
Steps to identify intersection points:
These points are important in solving a system of equations because they provide the solutions for both equations.
Steps to identify intersection points:
- First, graph both equations using a graphing calculator.
- Find where the two graphs intersect.
- The x and y coordinates of these points are solutions to the system of equations.
quadratic formula
The quadratic formula is a universal method for solving quadratic equations. It's given by: \(\bbox[yellow]{x = \frac{{-b 卤 u(b虏 - 4ac)}}{2a}}\)
This formula works for any quadratic equation of the form ax虏 + bx + c = 0. Here鈥檚 how you use it:
\(\bbox[yellow]{a = 3, b = 4, c = 1}\)
Applying the quadratic formula: \(\bbox[yellow]{x = \frac{{-4 卤 u(4虏 - 4路3路1)}}{2路3}}\)
This simplifies to: \(\bbox[yellow]{x = \frac{{-4 卤 u(16 - 12)}}{6}}\), resulting in: \(\bbox[yellow]{x = \frac{{-4 卤 2}}{6}}\)
This formula works for any quadratic equation of the form ax虏 + bx + c = 0. Here鈥檚 how you use it:
- Identify the coefficients a, b, and c from your equation.
- Plug these values into the quadratic formula.
- Simplify the expression to find the values of x.
\(\bbox[yellow]{a = 3, b = 4, c = 1}\)
Applying the quadratic formula: \(\bbox[yellow]{x = \frac{{-4 卤 u(4虏 - 4路3路1)}}{2路3}}\)
This simplifies to: \(\bbox[yellow]{x = \frac{{-4 卤 u(16 - 12)}}{6}}\), resulting in: \(\bbox[yellow]{x = \frac{{-4 卤 2}}{6}}\)
- Simplify further to find that x = -1 or x = -1/3.
