Respuesta :
Answer:
A) [tex]0.25x^2-0.6xy+0.36y^2=\left(0.5x-0.6y\right)^2[/tex]
B) [tex]-a^2+0.6a-0.09=-\left(10a-3\right)^2[/tex]
C) [tex]\frac{9a^4}{16}+a^3+\frac{4a^2}{9}=a^2(\left(9a+8\right)^2)[/tex]
D) [tex]-16m^2-24mn -9n^2=-\left(4m+3n\right)^2[/tex]
Step-by-step explanation:
The square of a binomial is the sum of: the square of the first terms, twice the product of the two terms, and the square of the last term.
[tex](a+b)^2 = a^2 + 2ab + b^2\\\\(a-b)^2 = a^2 - 2ab + b^2[/tex]
To find the square of the binomial of the following polynomials you must:
A) [tex]0.25x^2-0.6xy+0.36y^2[/tex]
Apply radical rule: [tex]a=\left(\sqrt{a}\right)^2[/tex]
[tex]0.25=\left(\sqrt{0.25}\right)^2\\0.36=\left(\sqrt{0.36}\right)^2[/tex]
[tex]\left(\sqrt{0.25}\right)^2x^2-0.6xy+\left(\sqrt{0.36}\right)^2y^2[/tex]
Apply exponent rule: [tex]a^mb^m=\left(ab\right)^m[/tex]
[tex]\left(\sqrt{0.25}\right)^2x^2=\left(\sqrt{0.25}x\right)^2\\\left(\sqrt{0.36}\right)^2y^2=\left(\sqrt{0.36}y\right)^2[/tex]
[tex]\left(\sqrt{0.25}x\right)^2-0.6xy+\left(\sqrt{0.36}y\right)^2[/tex]
Rewrite [tex]0.6xy[/tex] as [tex]2\cdot \:0.5x\cdot \:0.6y[/tex]
[tex]\left(\sqrt{0.25}x\right)^2-2\cdot \:0.5x\cdot \:0.6y+\left(\sqrt{0.36}y\right)^2[/tex]
Apply perfect square formula: [tex]\left(a-b\right)^2=a^2-2ab+b^2[/tex]
[tex]a=0.5x,\:b=0.6y[/tex]
[tex]\left(\sqrt{0.25}x\right)^2-2\cdot \:0.5x\cdot \:0.6y+\left(\sqrt{0.36}y\right)^2=\left(0.5x-0.6y\right)^2[/tex]
B) [tex]-a^2+0.6a-0.09[/tex]
Multiply both sides by 100
[tex]-a^2\cdot \:100+0.6a\cdot \:100-0.09\cdot \:100\\-100a^2+60a-9[/tex]
Factor out common term -1
[tex]-\left(100a^2-60a+9\right)[/tex]
Break the expression into groups and factor out common terms
[tex]-(\left(100a^2-30a\right)+\left(-30a+9\right))\\-(10a\left(10a-3\right)-3\left(10a-3\right))\\-(\left(10a-3\right)\left(10a-3\right))\\-\left(10a-3\right)^2[/tex]
C) [tex]\frac{9a^4}{16}+a^3+\frac{4a^2}{9}[/tex]
Apply exponent rule: [tex]a^{b+c}=a^ba^c[/tex]
[tex]a^3=aa^2\\a^4=a^2a^2[/tex]
[tex]\frac{9a^2a^2}{16}+aa^2+\frac{4a^2}{9}[/tex]
Factor out common term [tex]a^2[/tex]
[tex]a^2\left(\frac{9a^2}{16}+a+\frac{4}{9}\right)[/tex]
Factor [tex]\frac{9a^2}{16}+a+\frac{4}{9}\right[/tex]
Find the Least Common Multiplier (LCM) of 16, 9 which is 144.
Multiply by LCM
[tex]\frac{9a^2}{16}\cdot \:144+a\cdot \:144+\frac{4}{9}\cdot \:144\\81a^2+144a+64[/tex]
[tex]81a^2+144a+64=\left(9a\right)^2+2\cdot \:9a\cdot \:8+8^2[/tex]
Apply perfect square formula: [tex]\left(a+b\right)^2=a^2+2ab+b^2[/tex]
[tex]a=9a,\:b=8[/tex]
[tex]81a^2+144a+64=\left(9a+8\right)^2[/tex]
[tex]\frac{9a^4}{16}+a^3+\frac{4a^2}{9}=a^2(\left(9a+8\right)^2)[/tex]
D) [tex]-16m^2-24mn -9n^2[/tex]
Factor out common term -1
[tex]-\left(16m^2+24mn+9n^2\right)[/tex]
Break the expression into groups and factor out common terms
[tex]\left(16m^2+12mn\right)+\left(12mn+9n^2\right)\\4m\left(4m+3n\right)+3n\left(4m+3n\right)\\\left(4m+3n\right)\left(4m+3n\right)\\-\left(4m+3n\right)\left(4m+3n\right)\\-\left(4m+3n\right)^2[/tex]
[tex]-16m^2-24mn -9n^2=-\left(4m+3n\right)^2[/tex]
