a) Equation in factorized form: [tex](x+120)(x-60)=0[/tex]
b) Solutions: x = 60 and x = -120
c) Length of the box: x = 60 inches
Step-by-step explanation:
a)
The equation that represents the situation is:
[tex]x^2+60x-7200=0[/tex] (2)
We want to rewrite the equation in factored form, i.e. in the form
[tex](x+a)(x+b)=0[/tex] (1)
So we have to find the two numbers a, b.
We can note that developing the product in eq.(1), we find:
[tex](x+a)(x+b)=x^2+ax+bx+ab=x^2+(a+b)x+ab[/tex]
Which is also a quadratic equation, and by comparing with (2), we note that:
[tex]a+b=60\\ab=-7200[/tex]
So we have a system of two equations. We observe that both equations are satisfied if
[tex]a=120\\b=-60[/tex]
Therefore, the original equation in factorized form is
[tex](x+120)(x-60)=0[/tex]
b)
The zero product property states that if the product of two factors is equal to zero:
[tex]x_1 x_2 = 0[/tex]
Then either [tex]x_1[/tex] or [tex]x_2[/tex] must be zero.
In this problem, the factorized form of the equation is
[tex](x+120)(x-60)=0[/tex]
The two factors here are
[tex]x+120[/tex]
and
[tex]x-60[/tex]
According to the zero product property, the two solutions can be found by requiring the two factors to be zero. Therefore:
[tex]x+120=0 \rightarrow x=-120[/tex]
and
[tex]x-60=0 \rightarrow x=60[/tex]
So, these are the two solutions.
c)
Here we have to analyze the meaning of the equation in the context of the problem.
The problem says that the equation is used to find the length and the width of the base of the box, each measuring x inches.
We note that the length of the box must be a positive number (in fact, a negative length would have no meaning from a physical point of view).
This means that the value of x must be positive.
Of the two solutions found,
[tex]x=-120\\x=60[/tex]
Only the second one is positive: therefore, the only logical answer is
x = 60 inches
which is the length of the box.
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