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Chapter 8: Problem 61

For what value(s) of \(b\) does the following system of equations have two distinct, real solutions? $$\left\\{\begin{array}{l} y=-x^{2}+2 \\ y=x+b \end{array}\right.$$

Short Answer

Expert verified
The system of equation will have two distinct real solutions for any \(b\) that is greater than 1.

Step by step solution

01

Equating the Equations

The two given equations are \(y=-x^{2}+2\) and \(y=x+b\). Taxes are equalized: \(-x^{2}+2=x+b\)
02

Rearranging the Equations to Form Quadratic Equations

The equation \(-x^{2}+2=x+b\) is rearranged to form a Quadratic equation, it's given by:\(-x^{2}-x+2-b=0\)
03

Applying the Quadratic Formula

According to the quadratic formula, \(-b \pm \sqrt{b^{2}-4 a c}\) over \(2a\) provides the roots for the equation. In this case \(a=-1\), \(b=-1\), and \(c=2-b\). So the discriminant under the root which is \(b^2-4ac=(-1)^2-4 (-1)(2-b)\) must be positive for equation to have two distinct real solutions.
04

Evaluate the Discriminant

After simplification, the discriminant relation is given by:\(4-8+4b>0 => 4b>4 => b>1\)
05

Determine the Value of b

Based on our discriminant evaluation, this system has two distinct real solutions for \(b>1\). So the possible value(s) for \(b\) will be any number greater than 1.

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Key Concepts

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

System of Equations
A system of equations is a collection of two or more equations with shared variables. In our example, we have two equations involving the variables x and y:
  • \(y=-x^2 + 2\)
  • \(y = x + b\)
The goal is to find values of these variables that satisfy both equations simultaneously. By solving the system, you aim to find the points of intersection or solutions that work for both equations at the same time.
Systems of equations can be solved using different methods鈥攕uch as substitution, elimination, or graphical methods. In this exercise, we're focusing on substitution by equating the two expressions for y.
Solutions of Quadratics
When dealing with a quadratic equation, you're looking for solutions where the equation equals zero. Quadratic equations are of the form \(ax^2 + bx + c = 0\). The quadratic formula provides a way to find these values of x that solve the equation:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula is very useful as it gives the solutions directly, provided you know the values of a, b, and c.
For the system in question, after combining the equations into one, you need to rearrange it to fit the standard quadratic form. The solutions of the quadratic equation will represent the x-values where both original equations intersect. Howe many solutions there are is determined by the discriminant in the formula.
Discriminant
The discriminant in a quadratic equation is the part under the square root in the quadratic formula: \(b^2 - 4ac\). It plays a crucial role in determining the number and type of solutions:
  • A positive discriminant (\(b^2 - 4ac > 0\)) means there are two distinct real solutions.
  • A zero discriminant (\(b^2 - 4ac = 0\)) means there is exactly one real solution.
  • A negative discriminant (\(b^2 - 4ac < 0\)) means there are no real solutions, only complex ones.
In the exercise, the discriminant is evaluated to ensure it is positive, which results in the condition that \(b > 1\). Hence, for any b greater than 1, the quadratic equation will have two distinct real solutions, which is what we seek in the original system of equations.

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Most popular questions from this chapter

A manufacturer wants to make a can in the shape of a right circular cylinder with a volume of \(45 \pi\) cubic inches and a lateral surface area of \(30 \pi\) square inches. The lateral surface area includes only the area of the curved surface of the can, not the area of the flat (top and bottom) surfaces. Find the radius and height of the can.

If \(A=\left[\begin{array}{rrr}1 & 0 & 1 \\ 0 & 0 & 1 \\ 2 & -1 & 0\end{array}\right]\) and \(B=\left[\begin{array}{rrr}0 & 3 & -1 \\ -1 & 2 & 0 \\\ 0 & 0 & 1\end{array}\right],\) for what values of \(a\) and \(b\) does \(A B=\left[\begin{array}{rrr}0 & 2 a+2 b+1 & 0 \\ 3 a+4 b & 0 & 1 \\ 1 & 4 & -2\end{array}\right] ?\)

An airline charges 380 dollar for a round-trip flight from New York to Los Angeles if the ticket is purchased at least 7 days in advance of travel. Otherwise, the price is 700 dollar . If a total of 80 tickets are purchased at a total cost of 39,040 dollar, find the number of tickets sold at each price.

You wish to make a 1 -pound blend of two types of coffee, Kona and Java. The Kona costs \(\$ 8\) per pound and the Java costs \(\$ 5\) per pound. The blend will sell for \(\$ 7\) per pound. (a) Let \(k\) and \(j\) denote the amounts (in pounds) of Kona and Java, respectively, that go into making a 1 -pound blend. One equation that must be satisfied by \(k\) and \(j\) is $$k+j=1$$ Both \(k\) and \(j\) must be between 0 and \(1 .\) Why? (b) Using the variables \(k\) and \(j\), write an equation that expresses the fact that the total cost of 1 pound of the blend will be \(\$ 7\) (c) Solve the system of equations from parts (a) and (b), and interpret your solution. (d) To make a 1 -pound blend of Kona and Java that costs \(\$ 7.50\) per pound, which type of coffee would you use more of? Explain without solving any equations.

A farmer has 110 acres available for planting cucumbers and peanuts. The cost of seed per acre is \(\$ 5\) for cucumbers and \(\$ 6\) for peanuts. To harvest the crops, the farmer will need to hire some temporary help. It will cost the farmer \(\$ 30\) per acre to harvest the cucumbers and \(\$ 20\) per acre to harvest the peanuts. The farmer has \(\$ 300\) available for seed and \(\$ 1200\) available for labor. His profit is \(\$ 100\) per acre of cucumbers and \(\$ 125\) per acre of peanuts. How many acres of each crop should the farmer plant to maximize the profit?
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