Exercise 3 Page 515

We can solve any quadratic equation by completing the square. It is possible to use this method on a general quadratic equation to get a general formula. Let's give this a try. Recall that any quadratic equation can be written in standard form. ax^2+bx+c = 0 In this form a, b, and c are real numbers, and a ≠ 0. Remember that we can complete the square of an expression x^2+dx by adding ( d2 )^2. Therefore, our next step is to take the left-hand side of the equation to this form. Now, we can compare the left-hand side with x^2+dx to determine the constant that we need to add to complete the square. x^2+ dx x^2 + b/ax As we can see, d = ba. With this in mind we can proceed to find ( d2 )^2.

( d/2 )^2

Substitute

d= b/a

( ba/2 )^2

DivFracSL

.a/b /c.= a/b* c

( b/2a )^2

CalcPow

Calculate power

b^2/4a^2

Now let's add b^24a^2 to both sides of x^2 + bax = - ca to complete the square on the left-hand side. After that, we can factor the perfect square trinomial as the square of a binomial. This will allow us to solve the equation by taking the square root of both sides to isolate x afterwards.

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

AddEqn

LHS+b^2/4a^2=RHS+b^2/4a^2

x^2 + b/ax + b^2/4a^2 = b^2/4a^2 - c/a

FacPosPerfectSquare

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

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

ExpandFrac

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

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

Multiply

Multiply

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

SubFrac

Subtract fractions

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

SqrtEqn

sqrt(LHS)=sqrt(RHS)

x + b/2a = ± sqrt(b^2-4ac/4a^2)

SqrtQuot

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

x + b/2a = ± sqrt(b^2-4ac)/sqrt(4a^2)

CalcRoot

Calculate root

x + b/2a = ± sqrt(b^2-4ac)/2a

SubEqn

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

x = - b/2a ± sqrt(b^2-4ac)/2a

AddSubFrac

Add and subtract fractions

x = - b ± sqrt(b^2-4ac)/2a

We have obtained a formula that can be used to solve any quadratic equation. x = - b ± sqrt(b^2-4ac)/2a This important result is know as the Quadratic Formula.