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Big Ideas Math Geometry, 2014

BI

Big Ideas Math Geometry, 2014 View details

Maintaining Mathematical Proficiency

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Exercise 12 Page 527

To complete the square, make sure all the variable terms are on one side of the equation. Then, divide both sides of the equation by a so the coefficient of z^2 is 1.

z=1

Practice makes perfect

We want to solve the quadratic equation by completing the square. Note that all terms with z are on one side of the equation.

- z^2+ 2z=1

Now let's divide each side by - 1 so the coefficient of z^2 will be 1.

- z^2+ 2z=1

.LHS /- 1.=.RHS /- 1.

- z^2+ 2z/- 1=1/- 1

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Simplify left-hand side

Write as a sum of fractions

- z^2/-1+2z/-1=1/-1

MovePartNumRight

a* b/c=a/c* b

- 1/-1z^2+2/-1z=1/-1

Calculate quotient

z^2-2z=- 1

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

( b/2 )^2

b= - 2

( - 2/2 )^2

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Simplify

Calculate quotient

( - 1 )^2

Calculate power

1

Next, we will add ( b2 )^2=1 to both sides of our equation. Then, we will factor the trinomial on the left-hand side and solve the equation.

z^2-2z=- 1

LHS+1=RHS+1

z^2-2z+ 1 = - 1 + 1

FacNegPerfectSquare

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

(z-1)^2 = - 1 +1

Add terms

(z-1)^2 = 0

sqrt(LHS)=sqrt(RHS)

sqrt((z-1)^2)=sqrt(0)

Calculate root

z-1 = 0

LHS+1=RHS+1

z=1

The solution of the equation is z=1.

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Exercises p.527