completing the square and solving quadratics by taking square roots

Completing the Square and Solving Quadratics by Taking Square Roots

Perfect Square Trinomials•Some quadratics have special factoring

rules which we have not yet discussed•These are in the form a2-2ab+b2 or

a2+2ab+b2

•They can be simplified to (a-b)2 and (a+b)2

•They can also be simplified using the box method to get the same result

Factor•x2-10x+25•Matches which form? a2-2ab+b2 or

a2+2ab+b2

•Simplified form (a-b)2 and (a+b)2

Factor•x2+8x+16

Factor•40x=8x2+50

Why bother discussing perfect square trinomials now?•They show up in solving square roots!•They show up when we complete the

square!

Solving by taking square roots•To solve a quadratic equation you can

take the square root of both sides•First simplify so the squared term is on

one side of the equal sign and everything else is on the other!

•The key is to remember the positive and negative square roots

Solve 3x2-4

Solve 4x2-20=5

Solve x2-10x+25=27

Solve x2+8x+16=49

Completing the Square•Solving with the method above, can be

used only when the expressions work out to be nice squares

•When you are presented with a problem not modeled in the form x2+bx, you can add a term to form a perfect square trinomial

•Called completing the square

Completing the Square•To complete the square of x2+bx add

Example

•x2-2x + __________

•x2+4x+ __________

•x2+3x+ __________

•x2+5x+ __________

•x2-4x + __________

Steps in Completing the Square•1. Collect variable terms on one side of

the equation and constants on the other•2. As needed, divide both sides by “a” to

make the coefficient of the x2 term 2•3. Complete the square by adding (b/2)2

to EACH side of the equation•4. Factor the variable as a perfect square

trinomial•5. Take square root of both sides of the

equation and solve for the values of the variable including + and -

Solve X2=27-6x•1. Get all variables to one side

•2. Setup to complete the square

•3. Add (b/2)2 to each side

•4. Perfect square trinomial form

•5. Solve

Solve 2x2+8x=12•1. Get all variables to one side

•2. Setup to complete the square

•3. Add (b/2)2 to each side

•4. Perfect square trinomial form

•5. Solve

Solve x2-6x=-4•1. Get all variables to one side

•2. Setup to complete the square

•3. Add (b/2)2 to each side

Try in your groups•3x2-24x=27

•4. Perfect square trinomial form

•5. Solve