Module XIII, Lesson 3 Online Algebra VHS@pwcs Square Roots Module XIII, Lesson 3 Online Algebra VHS@pwcs

Square Roots If a2 = b, then a is the square root of b. For example, 62 = 36, so 6 is the square root of 36. We use the following symbol to mean square root. reads the square root of 25. Which is 5, 5 x 5 = 25 or -5, -5 x -5 = 25 The positive square root is what we call the principal square root. We use the symbol, +, to mean both the positive square root and the negative square root.

Find the square root of each number. 1.+ 2. - 3.+ Use a calculator 4. Find the square root of the numerator and denominator separately. 1. + 10 2. -12 3. +11.18033…. 4. 3/5

Rational Numbers Rational Numbers are numbers that can be written in the form where a and b are integers and b is not zero. These include: Whole numbers Integers Fractions Decimals – terminating (those that end) - Repeating ( those that have a pattern)

Irrational numbers Irrational numbers are numbers that can not be written as a fraction This includes decimals that have no pattern and never end. Pi is a good example of this. Its value is actually 3.141592654….. No pattern has ever been found nor has an end to the number been found.

Are the following numbers rational or irrational? 1. 2. 3. 4. Rational 7 Integers 2. Irrational 6.3245….. No pattern, does not repeat Rational 0.333333 Repeats Rational 2/3

Roots Another name that we sometimes use for solution is root. It is the place where the graph crosses the x-axis. We can graph the quadratics to find the solution or root.

Use your graphing calculator to graph the following equations Use your graphing calculator to graph the following equations. Note how many solutions each has and how many times it crosses the x-axis. y = x2 - 4 y = x2 y = x2 + 4 This graph crosses the x-axis twice, at (-2, 0) and (2, 0). This graph crosses the x-axis at one place, (0, 0) This graph never crosses the x-axis.

Remember the place where the graph crosses the x-axis is the root, or the solution. y = x2 - 4 y = x2 y = x2 + 4 This graph crosses the x-axis twice, at (-2, 0) and (2, 0). This graph crosses the x-axis at one place, (0, 0) This graph never crosses the x-axis. This means that the equation 0 = x2 – 4 has 2 solutions. x = 2 and x = -2 This equation has NO solution This equation 0 = x2 has 1 solution. x = 0

Solutions to Quadratic Equations. Quadratic equations can have: 1 solution 2 solutions No solution

Solving Simple Quadratics We can solve simple quadratics by: getting the x2 term by itself. Taking the square root of both sides Remember that your solution will be both the principal square root and and the negative square root.

The variable is already by itself, so go on to the next step. Solve the following. X2 = 64 Get the variable by itself. The variable is already by itself, so go on to the next step. Get the variable by itself, take the square root of each side.

Solve the following: 2x2 – 3 = 6 Get the variable by itself. 2x2 – 3 + 3 = 6 + 3 2x2 = 9 Take the square root of each side.

Solve the following: 2x2 + 16 = 0 Get the variable by itself. 2x2 + 16 = 0 2x2 + 16 – 16 = 0 – 16 2x2 = -16 Take the square root of both sides. This equation has no solution, since there is no number that we can multiply by itself to get -8.

So if you get x2 equal to a negative number, there is no solution. Remember A quadratic can have the following solutions. 1 solution 2 solutions No solution There is no number that you can multiply by itself that will give you a negative number. So if you get x2 equal to a negative number, there is no solution.