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Chapter 5: Q 63. (page 443)

Prove that every quadratic function of the form f(x)=x2+bx+ccan be written in the form fx=x-k2+C, with k=-b2andC=c-b24

Short Answer

Expert verified

The given statement is proved.

Step by step solution

01

Step 1. Given information

Quadratic function isf(x)=x2+bx+c

02

Step 2. Explanation

fx=x2+bx+c=x2+2b2x+b24+c-b24=x+b22+c-b24=x+k2+C

Here,

k=-b2,C=c-b24

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Most popular questions from this chapter

Solve the integral

鈭�4-xx-2dx

Solve the integral

鈭�1x2+9dx

Complete the square for each quadratic in Exercises 28鈥�33. Then describe the trigonometric substitution that would be appropriate if you were solving an integral that involved that quadratic.

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Solve the integral: 鈭�xlnx2dx

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The substitution x = 2 sec u is a suitable choice for solving鈭�1x2鈭�4dx.

(b) True or False: The substitution x = 2 sec u is a suitable choice for solving鈭�1x2鈭�4dx.

(c) True or False: The substitution x = 2 tan u is a suitable choice for solving鈭�1x2+4dx.

(d) True or False: The substitution x = 2 sin u is a suitable choice for solving鈭�x2+4鈭�5/2dx

(e) True or False: Trigonometric substitution is a useful strategy for solving any integral that involves an expression of the form x2鈭�a2.

(f) True or False: Trigonometric substitution doesn鈥檛 solve an integral; rather, it helps you rewrite integrals as ones that are easier to solve by other methods.

(g) True or False: When using trigonometric substitution with x=asinu, we must consider the cases x>a and x<-a separately.

(h) True or False: When using trigonometric substitution with x=asecu, we must consider the cases x>a and x<-a separately.
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