Solve each equation using the designated method

Question

Solve 2x^2 + 8x -4 = 0 by completing the square.

 

Solve (x^2 +x)^2 -3(x^2+x) +2=0 using substitution, factoring, and the quadratic formula

Answer 1

1. completing the square

 

  2x² + 8x - 4 = 0

  x² + 4x - 2 = 0

  x² + 4x = 2

 

add (1/2 * 4)² = 4 to both sides:

 x² + 4x + 4 = 2 + 4

 (x + 2)² = 6

 x + 2 = ± √6

 x = -2 ± √6

 

2.1 using substitution

 

assume y = (x² + x), replace (x² + x) with y in equation:

 

y² - 3y + 2 = 0

(y - 1)(y - 2) = 0

y = 1 or y = 2

 

y = x² + x = 1

or

y = x² + x = 2

 

use completing the square to solve the 1st equation:

 

x² + x = 1

x² + x + (1/2)² = 1 + (1/2)²

(x + 1/2)² = 5/4

x + 1/2 = ± √(5/4)

x = -1/2 ± √5/√4

x = -1/2 ± √5/2

x = (-1 ± √5)/2

 

You can use completing the square to solve the 2nd equation as well.

 

2.2 using factoring

 

(x² + x)² - 3(x² + x) + 2 = 0

(x2 + x - 1)(x2 + x - 2) = 0

 

x² + x - 1 = 0

or

x² + x - 2 = 0

 

the 1st equation is not easy to factor, the 2nd equation can be factored into

 

x² + x - 2 = 0

(x + 2)(x - 1) = 0

 

x = -2 or x = 1

 

2.3 using quadratic formula

 

(x² + x)² - 3(x² + x) + 2 = 0

 

to use quadratic formula, we still need to use substitution first, assume y = (x² + x), replace (x² + x) with y in equation:

y² - 3y + 2 = 0

 

use quadratic formula:

 

y = (-(-3) ±√((-3)² - 4 * 1 * 2)) / 2 * 1

   = (3  ± √(9 - 8)) / 2

   = (3 ± √1) / 2

 

y = (3 + 1) / 2 = 2

or

y = (3 - 1) / 2 = 1

 

then we have 

 

x² + x - 2 = 0

or

x² + x - 1 = 0

 

use quadratic formula to solve the 2nd equation:

 

x = (-1 ± √(1 - 4 * (-1))) / 2 

   = (-1 ± √5) / 2

  

Can you use quadratic formula to solve the 1st equation on your own?

 

 

Answer 2

2x² + 8x - 4 = 0

 

2(x + 2)² - 12 = 0

 

I obtained this form by considering the coefficient of x² , x, and the constant.  The coefficient of x² is 2, so I will put a 2 as the coefficient of the square.  Then I used the coefficient of x to determine my factor of the square.  Lastly, I expand the expression to get the last term.

 

Now we can solve for x.

 

2(x + 2)² = 12

 

(x + 2)² = 6

 

x + 2 = ± √(6)

 

x = -2 ± √(6)