We show the check for the first solution. The check for the other solution is similar.
CHECK Original equation
Let.Multiply.Simplify.✓ True
The solution set is , , abbreviated u.
3 322
2
v3 - 322
2
u v3 + 322
218 = 18
(^) A (^322) B
(^2)
18
(^) A 3 + 322 - (^3) B
(^2)
18
3 + 322c 2 a x= 2
3 + 322
2
b - 3 d
218
12 x- 322 = 18
SECTION 9.1 The Square Root Property and Completing the Square 499
NOW TRYThe symbol
denotes two
solutions.OBJECTIVE 4 Solve quadratic equations by completing the square.We can
use the square root property to solve anyquadratic equation by writing it in the form
Square of a binomial ConstantThat is, we must write the left side of the equation as a perfect square trinomial that
can be factored as the square of a binomial, and the right side must be a
constant. This process is called completing the square.
Recall that the perfect square trinomial
can be factored as
In the trinomial, the coefficient of x(the first-degree term) is 10 and the constant
term is 25. If we take half of 10 and square it, we get the constant term, 25.
Coefficient of x ConstantSimilarly, in
and in
This relationship is true in general and is the idea behind completing the square.
Solving a Quadratic Equation by Completing the SquareSolve
This quadratic equation cannot be solved by factoring, and it is not in the correct
form to solve using the square root property. To solve it by completing the square, we
need a perfect square trinomial on the left side of the equation.
Original equation
Subtract 10.We must add a constant to get a perfect square trinomial on the left.
Needs to be a perfect
square trinomialx^2 + 8 x+?
x^2 + 8 x=- 10
x^2 + 8 x+ 10 = 0
x^2 + 8 x+ 10 =0.
EXAMPLE 6 1 a 12c
1
2
1 - 6 2d
2m^2 - 6 m+ 9 , = 1 - 322 = 9.
c
1
2
112 2d
2x^2 + 12 x+ 36 , = 62 = 36 ,
c
1
2
110 2d
2= 52 = 25
x^2 + 10 x+ 25 1 x+ 522.
1 x+k 22 ,
1 x+k 22 =n.
⎧⎪⎪⎨⎪⎪⎩
NOW TRY
EXERCISE 5
Solve 15 x- 422 =27.
NOW TRY ANSWER
5.e
4 323
5
f