Question

Solving Quadratic Equations by completing the square:

z^2 - 3z - 5 = 0

Answer 1

Answer:
`z_1=4.19\\\\z_2=-1.19`

Explanation:

Add 5 to both sides of the equation:

`z^2 - 3z - 5 +5= 0+5\\\\z^2 - 3z = 5`

Divide the coefficient of
`z` by two and square it:

`((b)/(2))^2= ((3)/(2))^2`

Add it to both sides of the equation:

`z^(2) -3z+ ((3)/(2))^2=5+ ((3)/(2))^2`

Then, simplifying:

`(z- (3)/(2))^2=(29)/(4)`

Apply square root to both sides and solve for "z":

`\sqrt{(z- (3)/(2))^2}=\±\sqrt{(29)/(4) }\\\\z=\±\sqrt{(29)/(4)}+ (3)/(2)\\\\z_1=4.19\\\\z_2=-1.19`

Answer 2

`(z-(3)/(2) )^2-(29)/(4)`

Explanation:

We are given the following quadratic equation by completing the square:

`z^2 - 3z - 5 = 0`

Rewriting the equation in the form
`x^2+2ax+a^2` to get:

`z^2 - 3z - 5+(-(3)/(2) )^2-(-(3)/(2) )^2`

`z^2-3z+(-(3)/(2) )^2=(z-(3)/(2) )^2`

Completing the square to get:

`( z - \frac{ 3 } { 2 } )^ 2 - 5 - ( - \frac { 3 } { 2 } ) ^ 2`

`(z-(3)/(2) )^2-(29)/(4)`