Answer:
solution given:
A. 5x^2 + 15x + 11.25
taking common 5
5(x²+3x+11.25/5)
since 11.25/5=2.25=(3/2)^2
5(x²+2*x*3/2 +(3/2)²)
5(x+3/2)²
5(x+1.50)²
B.10x^2 +20x +10
taking common 10
10(x²+2x+1)
since x²+2x+1 is in a form of x²+2xy+y²
so
10(x+1)²
C. 1/4x^2 + x + 1
taking common 1/4
while taking common we must balance the equation
1/4(x² +4x+4)
since x²+4x+1 is in a form of x²+2xy+y²: x²+2*x*2 +2²
so
1/4(x+2)²
D. 3x^2 + 5x + 25/12
taking common 3
3(x²+5/3 x +25/(12*3))
3(x²+2* x*5/6+25/36)
since x²+2* x*5/6+25/36 is in a form of x²+2xy+y²: x²+2*x*5/6 +(5/6)²
so
3(x+5/6)²
Step-by-step explanation:
Answer:
Expand the two given forms:
[tex]\begin{aligned}\implies a(x+b)^2 & =a(x+b)(x+b)\\& =a(x^2+2bx+b^2)\end{aligned}[/tex]
[tex]\begin{aligned}\implies a(x-b)^2 & =a(x-b)(x-b)\\& =a(x^2-2bx+b^2)\end{aligned}[/tex]
Therefore, to write each given trinomial in one of the given forms:
[tex]\textsf{For }\quad 5x^2+15x+11.25:[/tex]
Factor out the coefficient of x² (common term 5):
[tex]\implies 5(x^2+3x+2.25)[/tex]
Factor the perfect trinomial:
[tex]\implies 5\left(x+\dfrac{3}{2}\right)^2[/tex]
[tex]\textsf{For }\quad 10x^2+20x+10:[/tex]
Factor out the coefficient of x² (common term 10):
[tex]\implies 10(x^2+2x+1)[/tex]
Factor the perfect trinomial:
[tex]\implies 10\left(x+1\right)^2[/tex]
[tex]\textsf{For }\quad \dfrac{1}{4}x^2+x+1:[/tex]
Factor out the coefficient of x² (common term 1/4):
[tex]\implies \dfrac{1}{4}(x^2+4x+4)[/tex]
Factor the perfect trinomial:
[tex]\implies \dfrac{1}{4}\left(x+2\right)^2[/tex]
[tex]\textsf{For }\quad 3x^2+5x+\dfrac{25}{12}:[/tex]
Factor out the coefficient of x² (common term 3):
[tex]\implies 3 \left(x^2+\dfrac{5}{3}x+\dfrac{25}{36} \right)[/tex]
Factor the perfect trinomial:
[tex]\implies 3\left(x+\dfrac{5}{6}\right)^2[/tex]
