Formula Formula A polynomial with degree 2 is called a quadratic polynomial. A quadratic equation can be simplified to the standard form: ax² + bx + c = 0 Where, a ≠ 0. A, b, c are coefficients. c is also called "constant". 'x' is the unknown quantity
Chapter 7, Problem 12CR
(a)
To determine
Tocalculate: The solution of the equation f(x)=0 where f(x)=2x2+3x+1 and g(x)=x2+3x+2 .
(a)
Expert Solution
Answer to Problem 12CR
Solution:
The solution set for the equation f(x)=0 is {−1,−12} .
Explanation of Solution
Given information:
The functions f(x)=2x2+3x+1 and g(x)=x2+3x+2 .
Formula used:
Quadratic formula for the zeros of the quadratic equationax2+bx+c=0
x=−b±b2−4ac2a .
Calculation:
Consider the equation f(x)=0 .
Solving f(x)=0 is equivalent to finding the zeros of the function f(x) using a quadratic formula.
By quadratic formula, the zeros of the quadratic equation 2x2+3x+1=0
x=−3±32−4⋅2⋅12⋅2=−3±9−84=−3±14 .
⇒x=−12orx=−1
Thus the solution set for the equation f(x)=0 is {−1,−12} .
(b)
To determine
Tocalculate: The solution of the equation f(x)=g(x) where f(x)=2x2+3x+1 and g(x)=x2+3x+2 .
(b)
Expert Solution
Answer to Problem 12CR
Solution:
The solution set for the equation f(x)=g(x) is {−1,1} .
Explanation of Solution
Given information:
The functions f(x)=2x2+3x+1 and g(x)=x2+3x+2 .
Formula used:
Quadratic formula for the zeros of the quadratic equation ax2+bx+c=0
x=−b±b2−4ac2a .
Calculation:
Consider the equation f(x)=g(x) ,
Solving f(x)=g(x) is equivalent to find the zeros of the function f(x)−g(x) using quadratic formula.
Now f(x)−g(x)=2x2+3x+1−(x2+3x+2)
⇒f(x)−g(x)=2x2+3x+1−x2−3x−2
⇒f(x)−g(x)=x2−1 .
By quadratic formula, the zeros of the quadratic equation x2−1=0
x=−0±02−4⋅1⋅(−1)2⋅1=±42=±22 .
⇒x=1orx=−1 .
Thus the solution set for the equation f(x)=g(x) is {−1,1} .
(c)
To determine
Todetermine: The solution of the inequality f(x)>0 where f(x)=2x2+3x+1 .
(c)
Expert Solution
Answer to Problem 12CR
Solution:
The inequality f(x)=2x2+3x+1>0 has the solution (−∞,−1)∪(−12,∞) .
Explanation of Solution
Given information:
Polynomials f(x)=2x2+3x+1 and g(x)=x2+3x+2 .
Consider the inequality f(x)>0 ,
From part (a), the real zeros of the function f(x)=2x2+3x+1 is x=−1,x=−12 .
Use the real zeroes to separate the real number line into three intervals as
(−∞,−1),(−1,−12),(−12,∞) .
To select a test point from each interval and determine whether f(x) is positive or negative.
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