College Algebra B Unit 8 Seminar Kojis J. Brown Square Root Property Completing the Square Quadratic Equation Discriminant

The Square Root Property If x 2 =a, then x = ±√[a] for all real numbers a Example: Solve x 2 =25 Answer, x = ±√[25] = ±5 (which means, x = 5 or x = -5)

Square Root Property Example Solve

Completing the Square This is one way to solve a quadratic equation Steps: 1) Put into form ax 2 +bx+c=0 2) Make sure a=1, if not-divide by “a” to make it so 3) Square half the coefficient of the linear term and add to both sides 4) Factor the left into a perfect square 5) Use the Square Root Property to solve

Completing the Square Example: Solve by completing the square x 2 +6x+8=0 1) Put into form ax 2 +bx=-cx 2 +6x=-8 2) Make sure a=1, if not-dividea=1 already by “a” to make it so 3) Square half the coefficient of the x 2 +6x +(6/2) 2 =-8 +(6/2) 2 linear term and add to both sidesx 2 +6x +(3) 2 = 1 4) Factor the left into a perfect square (x+3) 2 = 1 5) Use Square Root Property to solve (x+3) = ±√[1] = ±1 x+3 = 1 or x+3 = -1 x = -2 or x = -4 x = -4, -2

Completing the Square Here is another example. Solve by completing the square Remember the steps: 1) Put into form ax 2 +bx=-c 2) Make sure a=1, if not, divide by “a” to make it so 3) Square half the coefficient of the linear term and add to both sides 4) Factor the left into a perfect square 5) Use Square Root Property to solve

Quadratic Equation  Regardless of the method you are going to use to solve a quadratic equation, you must get the equation in STANDARD FORM.  The standard form of a quadratic equation has this format: ax 2 + bx + c = 0  a, b, and c are real numbers  a ≠ 0  Notice in STANDARD FORM  The highest exponent is TWO  There are no grouping symbols  All like terms are combined  Terms containing exponents are written such that the exponents are in descending order This is one of those pages you might want to keep handy!

Quadratic Equation  To use the QUADRATIC FORMULA to solve quadratic equations  Make sure the equation is in standard form  Identify the value of a, b, and c a = the coefficient of x 2, including the sign (if you see a negative) b = the coefficient of x, including the sign (if you see a negative) c = the constant (the plain old number), including the sign (if you see a negative)  Substitute these values into the quadratic formula  Perform the arithmetic This is another one of those pages you might want to keep handy!

Quadratic Formula For all equations ax 2 +bx+c=0,

 Solve 3x 2 – 10x + 3 = 0  Identify the value of a, b, and c a: b: c: Example Solving using the Quadratic Formula

 Solve 3x 2 – 10x + 3 = 0  Substitute these values into the quadratic formula Quadratic Equation

 Solve 3x 2 – 10x + 3 = 0  Perform the arithmetic

 Solve 3x 2 – 10x + 3 = 0  Perform the arithmetic Quadratic Equation

 So that we are all on the same page here … this means the equation 3x 2 – 10x + 3 = 0 has two solutions … x = 3 and x = 1/3 (Substituting either into the original equation will make the equation work out!) Quadratic Equation

 Solve x 2 + 9x = -1  Make sure the equation is in standard form Quadratic Equation x 2 + 9x + 1 =0

 Solve x 2 + 9x +1 = 0  Perform the arithmetic Quadratic Equation

 Solve x 2 + 9x +1 = 0  Perform the arithmetic Quadratic Equation

 So that we are all on the same page here … this means the equation x 2 + 9x + 1 = 0 has two solutions … x = (Substituting either into the original equation will make the equation work out!) Quadratic Equation

Discriminant In the Quadratic Formula, the expression b 2 -4ac is called the discriminant If the discriminant D is: The Quadratic Equation has: Positive 2 Real Solutions Zero 1 Real Solution Negative2 Complex Solutions

Discriminant Example: What type of solutions does the equation have? a)x 2 +6x+8=0 b) 5x 2 +10x+9=0 c) x 2 -8x=-16