Exercise 6 Page 586

We will explain how completing the square is used to derive the Quadratic Formula. We can consider a quadratic equation written in standard form, complete the square, and derive the formula by ourselves. Let's start by writing the constant term c on the right-hand side and dividing the equation by the leading coefficient a. Next, we want the left-hand side of the above equation to be a perfect square trinomial. For this to happen, we need three terms. The first and third terms must be perfect squares and the second term must be twice the product of the square root of the first and third terms. To achieve this, we will add ( b2a)^2 to both sides. Now we can rewrite the left-hand side to more clearly show that it is a perfect square trinomial. Then, we can factor it.

x^2+b/ax+(b/2a)^2=- c/a+(b/2a)^2

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Rewrite b/ax as 2x(b/2a)

ExpandFrac

a/b=a * 2/b * 2

x^2+2b/2ax+(b/2a)^2=- c/a+(b/2a)^2

MovePartNumLeft

a* b/c=a*b/c

x^2+2b/2ax+(b/2a)^2=- c/a+(b/2a)^2

CommutativePropMult

Commutative Property of Multiplication

x^2+2 x( b/2a)+( b/2a)^2=- c/a+(b/2a)^2

FacPosPerfectSquare

a^2+2ab+b^2=(a+b)^2

( x+ b/2a)^2=- c/a+(b/2a)^2

The left-hand side is already factored. The next step will be simplifying the right-hand side. We are almost there!

(x+b/2a)^2=- c/a+(b/2a)^2

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Simplify right-hand side

PowQuot

(a/b)^m=a^m/b^m

(x+b/2a)^2=- c/a+b^2/(2a)^2

PowProd

(a * b)^m=a^m* b^m

(x+b/2a)^2=- c/a+b^2/4a^2

ExpandFrac

a/b=a * 4a/b * 4a

(x+b/2a)^2=4a(- c)/4a^2+b^2/4a^2

MultPosNeg

a(- b)=- a * b

(x+b/2a)^2=- 4ac/4a^2+b^2/4a^2

AddFrac

Add fractions

(x+b/2a)^2=- 4ac+b^2/4a^2

CommutativePropAdd

Commutative Property of Addition

(x+b/2a)^2=b^2-4ac/4a^2

Finally, we will isolate the x-variable.

(x+b/2a)^2=b^2-4ac/4a^2

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Solve for x

SqrtEqn

sqrt(LHS)=sqrt(RHS)

x+b/2a=± sqrt(b^2-4ac/4a^2)

SubEqn

LHS-b/2a=RHS-b/2a

x=± sqrt(b^2-4ac/4a^2)-b/2a

CommutativePropAdd

Commutative Property of Addition

x=- b/2a± sqrt(b^2-4ac/4a^2)

MoveNegFracToNum

Put minus sign in numerator

x=- b/2a± sqrt(b^2-4ac/4a^2)

RootQuot

sqrt(a/b)=sqrt(a)/sqrt(b)

x=- b/2a± sqrt(b^2-4ac)/sqrt(4a^2)

RootProd

sqrt(a* b)=sqrt(a)*sqrt(b)

x=- b/2a± sqrt(b^2-4ac)/sqrt(4)sqrt(a^2)

CalcRoot

Calculate root

x=- b/2a± sqrt(b^2-4ac)/2sqrt(a^2)

Note that in one of the denominators we have the expression sqrt(a^2). Depending on the value of a, we can simplify this in two ways. sqrt(a^2) = a, ifa≥0 - a, ifa<0 However, since we are considering both the positive and negative signs for the term containing this expression, we can simplify sqrt(a^2) as a.

x=- b/2a± sqrt(b^2-4ac)/2sqrt(a^2)

SqrtPowToNumber

sqrt(a^2)=a

x=- b/2a± sqrt(b^2-4ac)/2a

AddSubFrac

Add and subtract fractions

x=- b± sqrt(b^2-4ac)/2a

Starting from a quadratic equation written in standard form, by completing the square, we have deduced the Quadratic Formula. ax^2+bx+c=0 ⇔ x=- b± sqrt(b^2-4ac)/2a