Question

How do you solve the quadratic equation by completing the square: `2x^2 - 7x = 2`?

Answer 1

The solution is `x = 7/4 +- (sqrt65)/4`.

Explanation:

`2(x^2 - 7/2x) = 2`
`x^2 - 7/2x + 49/16 = 1 + 49/16`
`(x - 7/4)^2 = 65/16`
`(x - 7/4) = +- (sqrt65)/4`

`x = 7/4 +- (sqrt65)/4`

Answer 2

`x=(7+sqrt65)/4,` `(7-sqrt65)/4`

Explanation:

`2x^2-7x=2`

Divide both sides by `2`.

`x^2-7/2x=1`

To complete the square means to force a perfect square trinomial on the left side of the equation in the form `a^2-2ab+b^2=(a-b)^2`.

Divide the coefficient of the `x` term by `2`, square the result, and add to both sides of the equation.

`(-7)/2-:2=(-7/2)*1/2=-7/4`
`(-7/4)^2=49/16`

`x^2-7/2x+49/16=1+49/16`

The common denominator for `1` and `49/16` is `16`. Multiply `1` times `16/16`, then add the two fractions.

`x^2-7/2x+49/16=16/16+49/16` =

`x^2-7/2x+49/16=65/16`

We now have a perfect square trinomial on the left side, where `a=x` and `b=7/4`.

`(x-7/4)^2=65/15`

Take the square root of both sides.

`x-7/4=+-sqrt(65/16)` =

`x-7/4=+-sqrt65/4`

Solve for `x`.

`x=7/4+-sqrt65/4` =

`x=(7+-sqrt65)/4`

`x=(7+sqrt65)/4` =

`x=(7-sqrt65)/4`