Question

Solve each quadratic equation by completing the square. 6. x2 + 2x = 8 7. x2 - 6x = 16 8. x2 - 18x = 19 9. x2 + 3x = 3 10. x2 + 7x = 8

Answer 1

Lets get started :)

These questions have asked us to solve by completing the square.
How do we? I have attached a picture, which will explain

6. x² + 2x = 8
→ b is the coefficient of x, which is 2
→ We take half of 2 and square it. Then, we add it to either side

x² + 2x
`+ ((2)/(2) )^2 = 8 + ( (2)/(2))^2`
x² + 2x + 1 = 8 + 1
( x + 1 )( x + 1 ) = 9
( x + 1 )² = 9

`√(( x + 1 )^2) = √(9)`
x + 1 = + 3 or x + 1 = - 3
x = 2 or x = - 4

7. x² - 6x = 16

→ We do the same thing we did in the previous question

x² - 6x +
`((6)/(2))^2 = 16 + ((6)/(2))^2`
x² - 6x + 9 = 16 + 9
(x - 3)² = 25

`√((x-3)^2) = √(25)`
x - 3 = + 5 or x - 3 = - 5
x = 8 or x = - 2

8. x² - 18x = 19

x² - 18x +
`( (18)/(2) )^2 = 19 + ( (18)/(2))^2`
x² - 18x + 81 = 19 + 81
( x - 9 )( x - 9 ) = 100
( x - 9 )² = 100

`√((x-9)^2) = √(100)`
x - 9 = + 10 or x - 9 = -10
x = 19 or x = - 1

9. x² + 3x = 3

x² + 3x +
`( (3)/(2) )^2 = 3 + ((3)/(2) )`

`x^2 + 3x + (9)/(4) = 3 + (9)/(4)`

`x^(2) + 3x + (9)/(4) = (21)/(4)`

`(x^2 + (3)/(2) ) ( x^2 + (3)/(2) ) = (21)/(4)`

`( x^2 + (3)/(2) )^2 = (21)/(4)`

`\sqrt{ x^2 + (3)/(2) } = \sqrt{ (21)/(4) }`
x +
`(3)/(2)` = +
`( √(21) )/(2)` or x +
`(3)/(2) = - ( √(21) )/(2)`
x =
`(-3+ √(21) )/(2)` or x =
`(-3 - √(21))/(2)`