Question

How do you complete the square to solve `2x² + 3x -5=0`?

Answer 1

`x = 1` or `x = -5/2`

Explanation:

You follow a series of steps to complete the square.

Step 1. Write your equation in standard form.

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

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

Add `5` to each side of the equation.

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

`2x^2+3x = 5`

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

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

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

`(3/2)^2/4 = (9/4)/4 = 9/16`

Step 5. Add the result to each side.

`x^2 +3/2x + 9/16 =5/2 + 9/16`

`x^2 +3/2x + 9/16= 40/16 + 9/16`

`x^2 +3/2x + 9/16 =49/16`

Step 6. Take the square root of each side.

`x+3/4 = ±7/4`

`x_1 + 3/4 = +7/4`

`x_1 = 7/4 – 3/4 = (7-3)/4 = 4/4 = 1`

`x_1 = 1`

`x_2 + 3/4 = - 7/4`

`x_2 = -7/4 – 3/4 = (-7-3)/4 = (-10)/4 = -5/2`

So `x = 1` or `x = -5/2`

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

(a) `x = 1`

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

(b) `x = -5/2`

`2x^2 + 3x -5 = 2(-5/2)^2 + 3(-5/2) -5 = 2(25/4) – 15/2 -5 = 25/2 -15/2-10/2 = (25-15-10)/2 = 0`

Answer 2

Solve: `y = 2x^2 + 3x - 5 = 0`

Explanation:

To solve this type of quadratic equations, use the Shortcut. It can save you a lot of work and time.
(a + b + c = 0) -> 2 real roots: 1 and `(c/a = - 5/2)`