Question

How do you complete the square to solve `0=5x^2 + 2x - 3`?

Answer 1

`x = 3/5` or `x = -1`

Explanation:

Step 1. Write your equation in standard form.

`5x^2 + 2x -3 = 0`

Step 2. Move the constant to the right hand side of the equation.

Add `3` to each side .

`5x^2+2x -3 +3 = 0+3`

`5x^2+2x = 3`

Step 3. Divide both sides of the equation by the coefficient of `x^2`.

Divide both sides by 5.

`x^2 +2/5x =3/5`

Step 4. Square the coefficient of x and divide by 4.

`(2/5)^2/4 = (4/25)/4 = 1/25`

Step 5. Add the result to each side.

`x^2 +2/5x + 1/25 =3/5 + 1/25`

`x^2 +2/5x + 1/25= 15/25 + 1/25`

`x^2 +2/5x + 1/25 =16/25`

Step 6. Take the square root of each side.

`x+1/5 = ±4/5`

Case 1

`x_1 + 1/5 = +4/5`

`x_1 = 4/5-1/5 = (4-1)/5`

`x_1 = 3/5`

Case 2

`x_2 + 1/5 = -4/5`

`x_2 = -4/5-1/5 = (-4-1)/5 = (-5)/5`

`x_2 = -1`

So `x = 3/5` or `x = -1`

Check: Substitute the values of `x` back into the quadratic.

(a) `x = 3/5`

`5x^2 + 2x -3 = 5(3/5)^2 + 2(3/5) -3 = 5(9/25) + 6/5 -3 = 9/5 +6/5 -15/5 = (9+6-15)/5 = 0`.

(b) `x = -1`

`5x^2 + 2x -3 = 5(-1)^2 + 2(-1) -3 = 5(1) – 2 -3 = 5-2-3 = 0`