COMPLETING THE SQUARE
IF WE TRY TO SOLVE this quadratic equation by factoring
x² + 6x + 2 = 0
we cannot. Therefore, we use a technique called completing the square. That means to make the quadratic into a perfect square trinomial, i.e. the form a² + 2ab + b² = (a + b)².
The technique is valid only when 1 is the coefficient of x².
1) Transpose the constant term to the right:
x² + 6x = −2
2) Add a square number to both sides. Add the square of half the coefficient of x. In this case, add the square of 3:
x² + 6x + 9 = −2 + 9
The left-hand side is now the perfect square of (x + 3).
(x + 3)² = 7.
3 is half of the coefficient 6.
This equation has the form
That is, the solutions to
x² + 6x + 2 = 0
are the conjugate pair,
−3 + , −3 − .
We can check this. The sum of those roots is −6, which is the negative of the coefficient of x. And the product of the roots is
(−3)² − ()² = 9 − 7 = 2,
which is the constant term. Thus both conditions on the roots are satisfied. These are the two roots of the quadratic.
Problem. Solve this quadratic equation by completing the square.
x² − 2x − 2 = 0
To see the solution, pass your mouse over the colored area.
See Lesson 37 of Algebra, Problems 6 and 7.
Before considering the quadratic formula, note that half of any
(Lesson 27 of Arithmetic, Question 4.)
The quadratic formula
ax² + bx + c = 0,
To prove this, we will complete the square. But to do that, the coefficient of x² must be 1. Therefore, we will divide both sides of the original equation by a:
This is the quadratic formula.
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