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A lesson from a high school algebra textbook that introduces the concepts of simplifying radicals and solving quadratic equations by completing the square. It includes exercises for students to practice these skills. The lesson covers the product of roots rule, perfect squares, and the process of completing the square to solve quadratic equations.
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Mastery Algebra 1 OBJ: Introduce simplifying radicals – part of Skill 17 9.5A
n the next two lessons, you will be solving quadratic equations by a couple of new methods that often produce solutions involving square roots. In Lesson 9-5A, you will learn a method for putting radicals into their simplest form, called simplifying radicals.
List the first 10 perfect squares: ___ , ___ , ___ , ___ , ___ , ___ , ___ , ___ , ___ , ___
Let’s review a little of what we know about radicals, or square roots. Find each root below:
Use your answers to the questions above to answer each question below:
These questions call attention to an important principle about how square roots work.
Use the principle above to find the answer to these questions involving roots:
This principle is most useful when we apply it in reverse. Instead of multiplying smaller roots to get one that is even larger and more difficult, we should try taking a large root and break it down into a product of smaller roots that we can do separately.
12 4 3 4 3 2 3 18 9 2 9 2 3 2 So 12 2 3 So 18 3 2
Product of Roots Rule The product of two roots is equal to the root of the product. a b a b
Mastery Algebra 1 OBJ: Introduce simplifying radicals – part of Skill 17 9.5B
We say that these answers are in simplest radical form. In other words, even though the numbers we started with were not perfect squares, we were able to find a perfect square that divided into them, and we did the square root of that number. The following box summarizes how this method for simplifying radicals works:
Example: Simplify 75 Step 1: 25 is the largest perfect square that divides into 75. (It goes in 3 times.) Step 2: 75 25 3 Step 3: 25 3 25 3 Step 4: 25 3 5 3 Therefore, 75 = 5 3
Use the process described above to simplify each radical below:
Simplifying Radicals by Perfect Squares
Step 1: Find the largest perfect square that divides into the number under the radical.
Step 2: Write the number under the radical as a product of this perfect square and another number.
Step 3: Write this radical of a product as the product of the two radicals.
Step 4: Do the square root of the perfect square and multiply this number by the remaining radical.
Mastery Algebra 1 OBJ: Introduce solving quadratic equations by completing the square 9.5D
he process of turning a polynomial into a perfect square is called completing the square. This idea can be used to help us solve quadratic equations, even ones that don’t factor. It works because when things are squared, we can undo the square by “unsquaring”, like we did in a previous activity.
Example 1: Solve x^2 2 x 7 0 Example 2: Solve j^2 14 j 6 5
Step 1: x^2 2 x 7 0 Step 1: j^2 14 j 6 5 +7 +7 +6 + x^2 2 x 7 j^2 14 j 11
Step 2: 22 1 , 12 1 , so we get x^2 2 x 7 Step 2: j^2 14 j 11 +1 +1 +49 + x^2 2 x 1 8 j^2 14 j 49 60 Step 3: x^2 2 x 1 8 Step 3: j^2 14 j 49 60 ( x 1 )^2 8 ( j 7 )^2 60
Step 4: ( x 1 )^2 8 Step 4: ( j 7 )^2 60
Step 5: x 1 2 2 2 Step 5: j 7 2 2 3 5 −1 −1 +7 + x 1 2 2 j 7 2 15
Use the process described above to solve each quadratic equation. Be sure to check your answers with the scrambled answers at the bottom of the next page.
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Solving quadratic equations by completing the square Step 1: Move the constant, if any, to the other side of the equation from the squared term. Step 2: Find half the coefficient of the linear term, square it, and add this value to both sides. Step 3: Complete the square. Step 4: “Unsquare” by taking the square root of both sides. Step 5: Solve for the variable.
Scrambled answers for 1-10:
Mastery Algebra 1 OBJ: Introduce solving quadratic equations by completing the square 9.5E