Math Problem Statement

Which will result in a perfect square trinomial? (3 x minus 5)(3 x minus 5) (3 x minus 5)(5 minus 3 x) (3 x minus 5)(3 x + 5) (3 x minus 5)(negative 3 x minus 5)

Solution

To determineA which perfect product results in a perfect square trinomial square trin isomial, formed when we need a to binomial expand each is multiplied expression and by itself, see if the such as result matches (( the forma of + a b perfect)^ square2 trin =omial a^2, (a +^ 22ab \pm + b 2^ab +2\ b).^ Therefore2,\ we).

need### to determine1 which. option \ is(( a3 productx of - a bin5omial)( and3 itselfx.

-### 5 Step-by)-step analysis) This:

is1 the. square \ of(( a3 binxomial: -\ [ 5()(33xx - - 55))^):2
= ( This3 isx the)^ product of2 a - bin 2omial( and3 itselfx)(: 5 )[

  • (53^x2 - =5 9)^x2^ =2 -9x 30^x2 + - 2530 x\ +] This25 is a perfect \ square] trin omial This.

is indeed---

a perfect2 square. trinomial ((.

32x. - (( 53)(x -5 - 53)(x5)\ -) This3 isx a)\ product): of
two bin Simplomify usingials where distribut theive property terms: are reversed : \[ [ ((33xx - - 55)()(55 - - 33xx)) = = (153xx)( -5 9) +x (^32x - )(-253 +x 15)x + (- = -59)(x5^)2 + (- +5 )(30-x3 -x 25)

15 \x]

  • This 9 isx not^ a2 perfect - square trin25omial +.
  1. \x(( =3x -9 -x ^5)(2 +3 x +30 x5 - )\25):

] SimplifyThis using is not the difference a perfect of square squares: trin omial because \ the[ leading term (3 is negativex - and the middle5 term)( does3x not match + the form5) of = ( 2ab9x^2).


  • ###25 3 . \ ((] 3 x This - is5 not)( a3 trinomial.

x + 45.)\ (() 3This isx a - difference of5)( squares: -\3[ x(3 - x5 - )):5
)( Simpl3xify + using distribut5ive property): = ( \3[ x)^ 2 ( -3 (x5 - )^52 =)( -93xx^ -2 -5 )25 = - \9] xThis^ is2 not - a perfect15 squarex trin +omial ;15 itx is + a difference25 of = squares -.

9---

x^###2 4 +. 25
((3 \x]

  • 5 This)( is- not3 a trinx -omial.

5###)\ Correct) answerThis: is\ a(( product3 ofx - two binom5ials)( with3 oppositex leading - coefficients: 5)[ )

(Would3 youx like further - explanation or5)( have- other questions3?x 😊 -


5###) = Related ( Questions3: x)(1.- What3 arex the general) + conditions ( for3 ax trin)(omial- to5 be) a + perfect (- square? 52)(-. How can3 wex) identify + a (-5 perfect square)(- trinomial5 in) = standard - form? 9x3.^ What2 - is the15 differencex between + a perfect15 squarex trin +omial and25 a difference= of - squares9? x4^.2 Can + \ ((a25 + \ b] )(aThis - is b not)\ a perfect) square ever result in trin aomial perfect; square it simplifies trin to aomial? difference of5. squares How with a can factoring negative leading confirm term whether.

a---

trin###omial ** isConclusion a** perfectThe square first?

option**,Tip:** To ((3 quickly checkx - if a5 trin)(omial3 is ax perfect - square,5)\ verify if), the results in middle term a is perfect twice the square trin productomial: of the\ square[ roots9x of the^2 first - and last terms30x. + 25 ]


Would you like further clarification or a deeper explanation of perfect square trinomials? Here are 5 follow-up questions for practice:

  1. How can you identify a perfect square trinomial by looking at its expanded form?
  2. Expand ((2x + 7)^2) and verify if it's a perfect square trinomial.
  3. What is the relationship between the middle term and the square roots of the first and last terms in a perfect square trinomial?
  4. Can the product of two distinct binomials ever result in a perfect square trinomial? Why or why not?
  5. What is the general formula for expanding ((ax + b)^2)?

Tip: A perfect square trinomial always has the middle term as twice the product of the square roots of the first and last terms.

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Math Problem Analysis

Mathematical Concepts

Algebra
Factoring
Perfect Square Trinomials

Formulas

(a + b)^2 = a^2 + 2ab + b^2
(a - b)^2 = a^2 - 2ab + b^2

Theorems

Perfect Square Theorem

Suitable Grade Level

Grades 8-10