See also: Quadratic Equation, Quadratic Equation Formula, numbers


Given a quadratic equation of the following form,

ax2 + bx + c = 0

we can find the roots by first converting the above form into the one below:

a ( x+p ) 2 + q = 0

where p = b 2a and q = c b2 4a .

The idea of completing the square comes from the fact that if we have a quadratic equation of the form:

(x+m) 2 = n

then it's easy to solve by taking the square root of both sides.

(x+m) 2 = ± n x+m = ± n x = m ± n

Follow the steps below to solve a quadratic equation in the general form by completing the square.

Original equation ax2 + bx + c = 0
Step 1. Divide the equation by a to get the coefficient of x2 equals to 1 x2 + bx a + c a = 0
Step 2. Move the constant term to the right hand side x2 + bx a = c a
Step 3. Add ( b 2a ) 2 to both sides of the equation x2 + bx a + ( b 2a ) 2 = c a + ( b 2a ) 2
Step 4. We can now re-write the left hand side as a complete square ( x + b 2a ) 2 = c a + ( b 2a ) 2
Step 5. Take the square root of both sides ( x + b 2a ) 2 = ± c a + ( b 2a ) 2 x + b 2a = ± c a + ( b 2a ) 2
Step 6. Move the constant term from left hand side to the right hand side, then solve for x x = b 2a ± c a + ( b 2a ) 2


Examples:

  1. Find the roots of the quadratic equation

    x2 4x + 3 = 0

    Step 1. We can skip this step since in this case a=1

    Step 2. Move the constant term to the right hand side

    x2 4x = 3

    Step 3. Add ( 42 ) 2 to both sides of the equation (i.e. add 4)

    x2 4x + 4 = 3 + 4 x2 4x + 4 = 1

    Step 4. Re-write left hand side as a complete square

    (x2) 2 = 1

    Step 5. Take the square root of both sides

    (x2) 2 = ± 1 x2 = ±1

    Step 6. Move the constant term from left hand side to the right hand side, then solve for x

    x = 2±1 x1 = 21 = 1 x2 = 2+1 = 3
  2. Find the roots of the quadratic equation

    x2 6x + 9 = 0

    Step 1. We can skip this step since in this case a=1

    Step 2. Move the constant term to the right hand side

    x2 6x = 9

    Step 3. Add ( 62 ) 2 to both sides of the equation (i.e. add 9)

    x2 6x + 9 = 9 + 9 x2 6x + 9 = 0

    Step 4. Re-write left hand side as a complete square

    (x3) 2 = 0

    Step 5. Take the square root of both sides

    (x3) 2 = ± 0 x3 = 0

    Step 6. Move the constant term from left hand side to the right hand side, then solve for x

    x = 3

    In this case, we could have skipped steps 2 & 3 too, because the original equation is already a complete square, i.e. we can straight away re-write the original equation as a complete square as shown in step 4.

  3. Find the roots of the quadratic equation

    2 x2 + 2x + 5 = 0

    Step 1. Divide the equation by 2

    x2 + x + 52 = 0

    Step 2. Move the constant term to the right hand side

    x2 + x = 52

    Step 3. Add ( 12 ) 2 to both sides of the equation (i.e. add 14)

    x2 + x + 14 = 52 + 14 x2 + x + 14 = 94

    Step 4. Re-write left hand side as a complete square

    ( x + 12 ) 2 = 94

    Step 5. Take the square root of both sides

    ( x + 12 ) 2 = ± 94 x + 12 = ± 94

    Step 6. Move the constant term from left hand side to the right hand side, then solve for x

    x = 12 ± 94 x1 = 12 3i 2 x2 = 12 + 3i 2

    This quadratic equation has 2 complex roots, as can be expected because the discriminant ( b2 4ac ) is negative.

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By Jimmy Sie

See also: Quadratic Equation, Quadratic Equation Formula, numbers