CHAPTER 13: QUADRATIC EQUATIONS AND APPLICATIONS …

1 CHAPTER 13 355 CHAPTER 13: QUADRATIC EQUATIONS AND APPLICATIONS CHAPTER Objectives By the end of this CHAPTER , students should be able to: Apply the square Root Property to solve QUADRATIC EQUATIONS Solve QUADRATIC EQUATIONS by completing the square and using the QUADRATIC Formula Solve APPLICATIONS by applying the QUADRATIC formula or completing the square Contents CHAPTER 13: QUADRATIC EQUATIONS AND APPLICATIONS .. 355 SECTION : THE square ROOT PROPERTY .. 356 A. SOLVE BASIC QUADRATIC EQUATIONS USING SQUAREROOT PROPERTY .. 356 B. ISOLATE THE SQUARED TERM .. 358 C. USE THE PERFECT square FORMULA .. 359 EXERCISE .. 360 SECTION : COMPLETING THE square .. 361 A. COMPLETE THE square .. 361 B. SOLVE QUADRATIC EQUATIONS BY COMPLETING THE square , a = 1 .. 362 C. SOLVE QUADRATIC EQUATIONS BY COMPLETING THE square , A 1 .. 363 EXERCISE .. 365 SECTION : QUADRATIC FORMULA.

2 366 A. DETERMINANT OF A QUADRATIC EQUATION .. 366 B. APPLY THE QUADRATIC FORMULA .. 368 C. MAKE ONE SIDE OF AN EQUATION EQUAL TO ZERO .. 370 EXERCISE .. 371 SECTION : APPLICATIONS WITH QUADRATIC EQUATIONS .. 372 A. PYTHAGOREAN THEOREM .. 372 B. PROJECTILE MOTION .. 373 C. COST AND REVENUE .. 374 EXERCISE .. 376 CHAPTER REVIEW .. 377 CHAPTER 13 356 We might recognize a QUADRATIC equation from the factoring CHAPTER as a trinomial equation. Although, it may seem that they are the same, but they aren t the same. Trinomial EQUATIONS are EQUATIONS with any three terms. These terms can be any three terms where the degree of each can vary. On the other hand, QUADRATIC EQUATIONS are EQUATIONS with specific degree on each term. Definition A QUADRATIC equation is a polynomial equation of the form + + = Where is called the leading term, is call the linear term, and is called the constant coefficient (or constant term).

3 Additionally, . SECTION : THE square ROOT PROPERTY A. SOLVE BASIC QUADRATIC EQUATIONS USING SQUAREROOT PROPERTY square root property Let and . Then = if and only if = In other words, = if and only if = or = MEDIA LESSON Solve basic QUADRATIC EQUATIONS using square root property (Duration 2:53) View the video lesson, take notes and complete the problems below Example: a) 8 2=648 b) 2=75 YOU TRY Solve. a) 2=81 b) 2=44 CHAPTER 13 357 MEDIA LESSON Solve EQUATIONS with even exponents (Duration 4:26) View the video lesson, take notes and complete the problems below Consider: 52= _____ and ( 5)2 = _____ When we clear an even exponent, we have _____.

4 Example: Solve. a) (5 1)2= 49 b) (3 +2)44=81 YOU TRY Solve. a) ( +4)2=25 b) (6 9)2=45 CHAPTER 13 358 B. ISOLATE THE SQUARED TERM Let s look at examples where the leading term, or squared term, is not isolated. Recall, the squared term must be isolated to apply the square root property. MEDIA LESSON Solve EQUATIONS using square root property Isolating the squared term 1st (Duration 5:00) View the video lesson, take notes and complete the problems below Before we can clear an exponent, it must first be _____. Example: a) 4 2(2 +1)2= 46 b) 5(3 2)2+6=46 YOU TRY Solve.

5 A) 5(3x 6)2+7=27 b) 5(r+4)2+1=626 Note: When we have the other side of the equation of a squared term negative, the equation does not have a real solution. For example, the equation 2= 1 does not have a real solution. There is a complex solution for this equation but we will not discuss it in this class. Example: Solve 2 2+5=4 2 2=4 5 2 2= 1 2= 12 This equation does not have a real solution. CHAPTER 13 359 C. USE THE PERFECT square FORMULA In order for us to be able to apply the square root property to solve a QUADRATIC equation, we cannot have the term in the middle because if we apply the square root property to the term, we will make the equation more complicated to solve. However sometimes, we have special cases that we can apply the perfect square formula to get rid of the term in the middle and then apply the square root property to solve the EQUATIONS .

6 Recall: Perfect square formula + + =( + ) or + =( ) MEDIA LESSON Solve EQUATIONS using square root property Perfect square formula (Duration 4:09) View the video lesson, take notes and complete the problems below Example: a) 2+8 +16=4 b) 9 2 12 +4=25 YOU TRY Solve. a) 2 6 +9=81 b) 9 2+30 +25=4 CHAPTER 13 360 EXERCISE Solve by applying the square root property. 1) ( 3)2=16 2) ( 2)2=49 3) ( 7)2=4 4) ( 5)2=16 5) ( +5)2=81 6) ( +3)2=4 7) ( +9)2=37 8) ( +5)2=57 9) ( 9)2=63 10) ( +1)2=125 11) (9 +1)2=9 12) (7 8)2=36 13) (3 6)2=25 14) 5( 7)2 6=369 15) 5( 5)2+13=103 16) 2 2+7=5 17) (2 +1)2=0 18) ( 4)2=25 19) 3 2+2 =2 +24 20) 8 2 29=25+2 2 21) 2( +9)2 19=37 22) 3( 3)2+2=164 23) 7(2 +6)2 5=170 24) 6(4 4)2 5=145 Apply the perfect square formula and solve the EQUATIONS by using the square root property.

7 25) 2+12 +36=49 26) 2+6 +9=2 27) 16 2 40 +25=16 28) 2+4 +4=1 29) 2 14 +49=9 30) 25 2+10 +1=49 CHAPTER 13 361 SECTION : COMPLETING THE square When solving QUADRATIC EQUATIONS previously (then known as trinomial EQUATIONS ), we factored to solve. However, recall, not all EQUATIONS are factorable. Consider the equation 2 2 7 =0. This equation is not factorable, but there are two solutions to this equation: 1 + 2 and 1 2. Looking at the form of these solutions, we obtained these types of solutions in the previous section while using the square root property. If we can obtain a perfect square , then we can apply the square root property and solve as usual. This method we use to obtain a perfect square is called completing the square . Recall. Special product formulas for perfect square trinomials: ( + ) = + + ( ) = + We use these formulas to help us solve by completing the square .

8 A. COMPLETE THE square We first begin with completing the square and rewriting the trinomial in factored form using the perfect square trinomial formulas. MEDIA LESSON Complete the square (Duration 5:00) View the video lesson, take notes and complete the problems below Complete the square . Find . + + is easily factored to _____ To make + + a perfect square , = _____ Example: a) 2+10 + b) 2 7 + c) 2 37 + d) 2+65 + CHAPTER 13 362 Note To complete the square of any trinomial, we always square half of the linear term s coefficient, , We usually use the second expression when the middle term s coefficient is a fraction. YOU TRY Complete the square by finding : a) 2+8 + b) 2 7 + c) 2+53 + B.

9 SOLVE QUADRATIC EQUATIONS BY COMPLETING THE square , a = 1 Steps to solving QUADRATIC EQUATIONS by completing the square Given a QUADRATIC equation + + = , we can use the following method to solve for . Step 1. Rewrite the QUADRATIC equation so that the coefficient of the leading term is one, and the original constant coefficient is on the opposite side of the equal sign from the leading and linear terms. + _____ = +_____ Step 2. If , divide both sides of the equation by Step 3. Complete the square , , and add the result to both sides of the QUADRATIC equation. Step 4. Rewrite the perfect square trinomial in factored form. Step 5. Solve using the square root property. Step 6. Verify the solution(s). MEDIA LESSON Solve QUADRATIC equation by completing the square (Duration 8:40) View the video lesson, take notes and complete the problems below Solve the QUADRATIC equation using the square root principle.

10 ( 5)2=28 CHAPTER 13 363 Example: a) 2+6 9=0 b) 2 5 +10=0 c) 3 2+2 9=0 YOU TRY Solve. a) 2+10 +24=0 b) 2 8 +4=0 C. SOLVE QUADRATIC EQUATIONS BY COMPLETING THE square , A 1 MEDIA LESSON Solve QUADRATIC equation by completing the square a 1 (Duration 4:59) View the video lesson, take notes and complete the problems below To complete the square : 2+ + =0 1. Separate _____ and _____ 2. Divide by _____ (everything!) 3. Find _____ and _____ to _____ CHAPTER 13 364 Example a) 3 2 15 +18=0 b) 8 +32=4 2 YOU TRY Solve.