Exercise 65 Page 139

We want to solve the quadratic equation by completing the square. Note that all terms with x are on one side of the equation. To do so, we will start by rewriting the equation so all terms with x are on one side of the equation. 4x^2=20x - 25 ⇔ 4x^2 - 20x = - 25 Now let's divide each side by 4 so the coefficient of x^2 will be 1.

4x^2 - 20x = - 25

DivEqn

.LHS /4.=.RHS /4.

4x^2 - 20x/4=- 25/4

▼

Simplify left-hand side

WriteDiffFrac

Write as a difference of fractions

4x^2/4-20x/4=- 25/4

MovePartNumRight

a* b/c=a/c* b

4/4x^2-20/4x=- 25/4

CalcQuot

Calculate quotient

x^2-5x=-25/4

MoveNegNumToFrac

Put minus sign in front of fraction

x^2-5x=- 25/4

In a quadratic expression, b is the linear coefficient. For the equation above, we have that b=- 5. Let's now calculate ( b2 )^2.

( b/2 )^2

Substitute

b= - 5

( - 5/2 )^2

▼

Simplify

MoveNegNumToFrac

Put minus sign in front of fraction

(- 5/2 )^2

NegBaseToPosBase

(- a)^2=a^2

(5/2 )^2

PowQuot

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

5^2/2^2

CalcPow

Calculate power

25/4

Next, we will add ( b2 )^2= 254 to both sides of our equation. Then, we will factor the trinomial on the left-hand side, and solve the equation.

x^2-5x=- 25/4

AddEqn

LHS+25/4=RHS+25/4

x^2-5x + 25/4=- 25/4 + 25/4

FacNegPerfectSquare

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

(x-5/2)^2=- 25/4+24/4

AddFrac

Add fractions

(x-5/2)^2=0

SqrtEqn

sqrt(LHS)=sqrt(RHS)

sqrt((x-5/2)^2)=sqrt(0)

CalcRoot

Calculate root

x-5/2=0

AddEqn

LHS+5/2=RHS+5/2

x=5/2