Mathematics Notes by Success Tutorials: Solving Quadratic Equations
Success.lt
Completing the Square to Solve Quadratic Equations
Overview: This is a method to solve quadratic equations by creating a perfect square and a numerical remainder. It is also used to graph parabolas and quickly determine the vertex, maximum or minimum and direction (up or down).
Method: i.e. 4x2 + 8x + 19 = 31
1. Change the co-efficient of x2 to 1 by multiplying both sides of the equation by 1 over the co-efficient of x2.
= 1/4(4x2 + 8x + 19) = 1/4(31)
= 1/4(4x2) + 1/4(8x) + 1/4(19) = 1/4(31)
= x2 + 2x + 19/4 = 31/4
2. Move all non-x2 and non-x terms to one side of the equation.
= x2 + 2x + 19/4 - 19/4 = 31/4 - 19/4
= x2 + 2x = 12/4
= x2 + 2x = 3
3. Create a quadratic which can be factored into a perfect square on the side with the x2 term.
= x2 + 2x + 1 = 3 + 1 note: add the square of 1/2 of the co-efficient of the x term to both sides:
1/2(2) = 1, 1 x 1 = 1
= x2 + 2x + 1 = 4
4. Factor the quadratic into a perfect square.
= (x + 1)(x + 1) = 4
= (x + 1)2 = 4
5. Take the square root of both sides.
= (x + 1) = +/-2 note: 2 x 2 = 4 and -2 x -2 = 4, you need to use both the positive and negative square roots
6. Solve the equation, taking into account the +/- square root, which will give two answers.
= x + 1 = 2
= x = 1
or = x + 1 = -2
= x = -3
therefore, x = 1 or -3
Opdateret d. 23/2/01
