Solve the quadratic equation by completing the square

Question

x² + 3x - 6 = 0 

Answer 1

Hi Tarryn, to complete the square, first find the value of (b/2)² were b is the coefficient of the x term. The coefficient of the x term is 3, then the value of (b/2)² is (3/2)².

 

To complete the square start by rewriting the equation as:

 

x² + 3x = 6

 

Next add (3/2)² to both sides of the equation:

 

x² + 3x + (3/2)² = 6 + (3/2)²

 

The right side becomes:

 

(x + 3/2)² 

 

The left side becomes:

 

33/4

 

The new equation:

 

(x + 3/2)² = 33/4

 

Take the square root of both sides:

 

x + 3/2 = ±√(33/4)

 

Subtract 3/2 from both sides:

 

x = -3/2 ±√(33/4)

 

We can rewrite this as:

 

x = -3/2 ±(√33)/2, since the square root of 4 is 2.

 

Finally, we can rewrite this as:

 

x = (-3 ±√33) / 2

 

The solutions are:

 

x = (-3 +√33) / 2 and x = (-3-√33) / 2

 

Note: You get the same answers if you use the quadratic formula. This is a good check on your completing the square answers.

 

Questions?

Answer 2

x^2 + 3x - 6 = 0

 

to the left side of the equation; add and subtract (1/2 the coefficient of x)^2

 

x^2 + 3x  + (3/2)^2 -(3/2)^2 - 6 = 0

 

[x^2 + 3x + (3/2)^2] -(3/2)^2 - 6 = 0

 

(x + 3/2)^2  = - 9/4 - 6 = -8 1/4 = -33/4

 

taking square roots on both sides

 

x + 3/2 = +-(33)^1/2/2

 

x =(-3 +-(33)^1/2)/2

 

x = (-3 +(33)^1/2)/2 OR  (-3 -(33)^1/2)/2

 

x = +1.37 or -4.37

 

Checking (1.37)^2 + 3 * 1.37 -6 = (-4.37)^2 + 3*(- 4.37) - 6 = -0.01 = 0 (approx)