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Maths Exercices

Solve a quadratic equation by completing the square


With quadratic equations (ax2 + bx + c = 0), you can solve by completing the square.

Step 1: Make sure that the left side of the equation looks like x2 + bx.

Step 2: Add to both sides.

Step 3: Factor the left side as

Step 4: Take the square root and solve.


Solve by completing the square.

x2 - 2x - 3 = 0
Step 1: Make sure that the left side of the equation looks like x2 + bx.

To make the left side of the equation look like x2 + bx, add 3 to both sides.

x2 - 2x - 3 = 0
     
x2 - 2x = 3


Step 2: Add to both sides.

Since b=-2,

Add 1 to both sides.

x2 - 2x = 3
     
x2 - 2x + 1 = 4

Step 3: Factor the left side as

In general, an expression of the form can be factored as

The expression x2 - 2x + 1 is of this form, with b=-2. So, it can be factored as (x-1)2.

Rewrite the equation with the left side factored.

(x-1)2 = 4

Step 4: Take the square root and solve.

x - 1 =   Take the square root
         
x =   Add 1 to both sides
         
x = 1+2   or   x = 1-2         Split into + or -
         
x=3 or x=-1   Simplify

Solve for by completing the square

w2+5w-24 = 0
w = or w =