PARALLEL $TFs1$ (Performance $Tost\right)$ Answer the following problems. (5pts. $E.acl1\right)$ $1.$ The area of a square is $x^{2}+8x$ $16$ square units. What expression represents the + length of the side? $2$ The area of a rectanguiar thalkboard $dis\left(x^{2}$ $14x+45\right)m^{2}$ what are $11s$ dimensions? $3.$ If the area of a square garden $is\left(4x2-12x+9\right)$ square units, is it possible to solve its * sides? $A$ $cs$ using factoring difference of two squares. $3$ $N0$ one of the sides must be given $C$ Yes, the area is a perfect square trinomial $D$ $N0$ the area is not factorable

x = 3 and x =- 5 Like $\square commen$ INTERACTIVE PHASE complePtinng d the the solutions of each of the following quadratic equation by square. - $1$ $x^{2}-4x$ $32=0$ $24x^{2}-3\bar{2x} +28=$ 3. x - 8x + 15=0 ENGAGING - Let's Get Ready Solve each equation using the completing the square method. 1x+ 3x - 40 =0 2. x + 10x = 3 3. x - 3x – 10 - 0 REFLECTION - Let's Wrap-up! + How did vou find the solutions of each equation? + What mathematies concepts or principles did you apply in finding the solutions? Like $\square commen$ Lesson 2.d: Solves quadratic equations by using the quadratic formula. Concept: +The quadratic formula can be used to solve all types of quadratic equation. +To use the quadratic formula, the equation should be in general form which is, ax2 + bx +c=0 the equation, ax? is the quadratic term, bx is the linear term + In and c is the constant term. +The quadratic formula is Review: Write the following quadratic equation in standard fom, ax+bx+-0, then identify the values of a, b and e. 1. 2x + 9x-10 2. 2x(x - 6) =0 3. (x+ 4Xx + 12) -0 $-x+1=0$ $a$ $-$ $1$ $x$ - - O Like $\square commen$ MODELLING * The derivation of the quadratic formula uses the completing the square method. 1) ax? + bx +c = 0 Add - to both sides, simplify 2) ax? + bx = -c Divide both sides by a, 3) x2 bx simplify a a bx b2 b2 complete the square on the left 4) x2 + a 4a2 side of the equation by adding the constant to b2 both side of the equation 5) (z+ %3D a 4a2 62-tac factor the perfect square 6) (x+2) - trinomial 4a2 simplify b2-4ac get the square root of both 7) 4a2 sides +Vb2-4ac 8) x+ %3D 2a 2a simplify ±VB2-4ac 9) x = add to both sides to 2a 2a -b+Vb2-4ac isolate x 10) x = 2a combine the two fraction; this is the quadratic formula Examples: 1. Find the solution of the equation 2x+ 3x - 27 = 0 using the quadratic formula. Solution: 2x+ 3x - 27 = 0 determine the values of a, b and c a = 2; b = 3; c = -27 -313-4(2)(-27) 2(2) substitute the values of a, b and c in the quadratic formula Like Comment -3+V94216 simplify -3+/225 solve for x X= -3 3+15 *=3-15 D the solutions are x = 3 and x= - 2. Find the solution of the equation x - 4x - 6 = 0 using the quadratic formula. Solution: *-4x - 6 = 0 determine the values of a, bandc a = 1; b=-4; c =-6 -(-4)t -4(1)(-6) substitute the values of a, b and cin 2(1) the quadratic formula simplify ) since v40 = V4 VI0 = 2/10 xx= = 2 2++ VV1I0O ; x 2 is a factor of every tem, thus we 2 csaon lve difvior de x out abre y thx e =s2ol-utVio1n0 s = 2-V10 D x = 2+y10 and Reminder: the • -Ib f formb utilha as nt ies gapapotpsieivtaei, vres t. hien n the O Like $\square commen$ PHASE $8n$ $→n$ $1$ solutions of each of the following equations using the xx x* - + + I411Nx T00x x -ER8-AC0 TIVE formula. + + 921 =th0 0 e Find ENGAGING - Let's Get Ready Direction: Use the quadratic formula to find the solutions of each of the following equations. 1. x- 7x + 10 0 2. x+ 8x + 12 -0 3. 2x-6x+ 1= 0 REFLECTION - Let's Wrap-up! What happens if in the quadratic equation gives a negative answer? Can you explain why? 1 Like $\square commen$ O Lesson 3: The nature of the Roots of a Quadratic Equation Concept: +The value of the expression b? - 4ac is called the discriminant of the quadratic equation ax? + bx +c =0. +This value can be used to describe the nature of the roots of a quadratic equation. It can be zero, positive and perfect square, positive but not perfect square or negative. Review: Math in A, B C? Write the following quadratic equations in standard fom, ax+bx+e-0, then identify the values of a, b and e. 1. x= 8x - 3 2,+ Sx-4 534.. . + -(2x 8-xx 1- 4=3) 4 = x=12 0 MODELING When b- 4ac is equal to zero, then the roots are real mumbers and are equal. Example: Describe the roots of x-4x + 4-0 Solution: x- 4x + 4-0 detemine the values of a, b and e a = 1; b=;c=4 b- 4ạc = - 4(1)(4) substitute the values of a, b and e in the expression b- 4ae 2 Like $\square commen$ = 16-16 simplify =0 the value of b 4ac is zero, the roots are real mber and are equal 2. When b- 4ac is greater than zero and a perfect square, then the roots are rational numbers but are not equal. Example: Detemine the nature of the roots of x + 7x + 10-0 Solution: x+ 7x + 10-0 detemine the values of a, b and e substitute the values of a, b and e in $b=7$ $=10$ $4x-n-$ $0$ the expression b- 4ac - simplify the value of b- 4ac is greater than zero and a perfect square, the roots are rational mumber but are not equal 3. When b- 4ac is greater than zero but not a perfect square, then the roots are irrational umbers and are not equal. Exanple: 6x + 3 =0 Solution: roots $Dosnbeaetosat$ $x^{2}+6x+3$ $0$ $=1:b-6e=3$ $b^{2}-4\infty -62-4\left(1\right)\left(3$ c = 3 of x+ detemine the values of a, b and e substitute the values of a, b and e in the expression b- 4ac = 36 - 12 simplify = 24 the value of b-4ac is greater than zero but not a perfect square, the roots are irrational and are not equal 4. When b- 4ac is less than zero, then the equation has no real solution. Example: Detemine the nature of the roots of x? + 2x + 5=0 Solution: x + 2x + 5-0 detemine the values of a, b and e a = 1; b=2; c=5 b2- 4ac - 22-4(1)(5) substitute the values of a, b and e in the expression b-4ac =4- 20 simplify 3 O Like $\square commen$ the value of b- 4ac is less than zero and has no real solution PHASE of the roots of the following quadratic equations using $\dfrac {8m} {1-}$ $\infty $ Determine + - I843x x Nx T+ + -E1316 R-- tA-ho e 0 C0 nTaItVure E ENGAGING -Let's Get Ready What is My Nature? Direction: Describe the roots of the following quadratic equations using the discriminant. 1. 4x - 4x + 1 -0 2. 2x - 6x + 2 =0 3. 2x - 10x + 8-0 REFLECTION -Let's Wrap-up! Answer the following questions + How do you detennine the nature of the roots of a quadratic equation? + Karen says that the quadratic equation 2x2 + 5x - 4 = 0 has two possible solutions because the value of its discriminant is positive. Do you agree with Karen? Justify your answer. $BLike$ $\square comment$ Lesson 4: Solves equations transformable to quadratic equation Concept: + There are equations that are transformable into quadratic equation. These equations may be given in different form. + Once the equations are transformed into quadratics, they can be solved using the different methods of solving quadratic equations, such as extracting square roots, factoring, completing the square and using the quadratic formula. + An extraneous root or solution is a solution of an equation derived from an original equation. However, it is not a solution of the original equation. Review: A. Let's Recall Direction: Find the solution's of the following quadratic equation. Use ay method. 1. x- 4x + 4= 0 2. x+ 12x - 28 -0 3. x+ 4x - 32 -0 and Subtract Perform the indicated operation then express your simplest form. 3B21. . . . + a-4 LDenits's rw2ecex tr ion: Add - in 1 O Like $\square commen$ MODELING A. Solving Quadratic Equations that are Not Written in Standard Form Example: Solve x(x - 5) = 36 Solution: X(x - 5) = 36 not written in standard form x- 5x = 36 simplify using the Distributive Property of Multiplication x - 5x - 36 =0 write in standard form (x - 9)(x + 4) =0 factor the equation x-9-0 x+4-0 solve for x x=9 > the roots are x=9 and x=-4 B. Solving Rational Algebraic Expressions Transformable into Quadratic Equation Example: 1. Solve the rational algebraic equaiton+ - 2 Solution: 4x( +) - 4x(2). multiply both sides by the least common multiple of all denominators, in this equation the LCM is 4x = 8x x*24 -- + 31x1x x - + 832x x 4 + 24 = - = 0 $\bar{k} $ ssciomolmpvbe liiin ftfny he osr e ttx aeenrqdmas uartd iofn orm (x- 8)(x- 3) = 0 $-$ $-$ $-$ x-8 = 0 X-3 X=8 X=3 the solutions are x=8 and x=3 2. Find the roots of x +=1+

Il Solve the following $0roblcmi$ Write the correct answer on the space before each number (SHOW YOUR SOLUTIONS) Please utilize the space at the back of your test paper for your $5olutlon$ 41. A rectangular garden in a backyard has an area $of\left(3x^{2}+5x-6\right)$ square meters. Its width is $\left(x+2\right)metcrs$ Find the length of the $g0rden$ 42. The side of a square lot is $\left(5x-3\right)mctcr5$ $HOwmanymeter5Offcncing$ $mater1a15$ Are needed to enclose the square lot? expression represents 43. If a square has a perimeter $o1\left(4\times -96\right)meter5,Mhate\times PFC$ $2nt5the$ Area of the square? 44. Find all the real roots of the polynomial equation $x^{3}-6x^{2}+11x-6=0$ 45. Determine the number of real roots of the polynomial equation $x^{2}\left(x^{3}-1\right)=0$ 46. The roots of the polynomial equation $are2,^{.2},7and.7,$ $Crcatcthe$ polynomial Equation using zero product property. 47. The factored form of a polynomial The real roots of the polynomial equation? $\left(x-2\right)\left(3\times +3\right)\left(x-3\right)$ $whatarc$ $hor\left(x-1\right)l5atactor0t\left(x^{3}-\times -2\right)$ $Mhat|$ 48. Use factor theorem to determine whether The remainder of the two polynomials? 49. The volume of a rectangular solid is $\left(2x^{3}+3x^{2}+2x-5\right)cdb|ccm,and|tsh$ height is (x+1) cm. What is the area of the solid? 50. The real roots of the polynomial equation x'+ $2x^{2}-23x-60=0are-4,-3and5$ Calculate the other possible rational numbers that will satisfied the equation.