Answer:
Q1 - x = -2 and [tex]x=\frac{1}{2}[/tex]
Q2 - [tex]x=\frac{9}{4}[/tex] and [tex]x=\frac{-3}{2}[/tex]
Q3 - Square, Length = 19 m
Step-by-step explanation:
Question 1:
We have the equation [tex]2x^{2}+3x=2[/tex] i.e. [tex]2x^{2}+3x-2=0[/tex]
The solution of equation [tex]ax^{2}+bx+c=0[/tex] is [tex]x=\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}[/tex].
So, from the given equation, we get,
a = 2, b = 3 and c = -2.
Thus, [tex]x=\frac{-3\pm \sqrt{(3)^{2}-4\times 2\times (-2)}}{2\times 2}[/tex]
i.e. [tex]x=\frac{-3\pm \sqrt{9+16}}{4}[/tex]
i.e. [tex]x=\frac{-3\pm \sqrt{25}}{4}[/tex]
i.e. [tex]x=\frac{-3\pm 5}{4}[/tex]
i.e. [tex]x=\frac{-3+5}{4}[/tex] and [tex]x=\frac{-3-5}{4}[/tex]
i.e. [tex]x=\frac{2}{4}[/tex] and [tex]x=\frac{-8}{4}[/tex]
i.e. [tex]x=\frac{1}{2}[/tex] and x= -2
Question 2:
We have the equation [tex]16x^{2}-12x=54[/tex] i.e. [tex]16x^{2}-12x-54=0[/tex]
Again, we have,
a = 16, b = -12 and c = -54
Thus, [tex]x=\frac{12\pm \sqrt{(-12)^{2}-4\times 16\times (-54)}}{2\times 16}[/tex]
i.e. [tex]x=\frac{12\pm \sqrt{144+3456}}{32}[/tex]
i.e. [tex]x=\frac{12\pm \sqrt{3600}}{32}[/tex]
i.e. [tex]x=\frac{12\pm 60}{32}[/tex]
i.e. [tex]x=\frac{12+60}{32}[/tex] and [tex]x=\frac{12-60}{32}[/tex]
i.e. [tex]x=\frac{72}{32}[/tex] and [tex]x=\frac{-48}{32}[/tex]
i.e. [tex]x=\frac{9}{4}[/tex] and [tex]x=\frac{-3}{2}[/tex]
Question 3:
We have that,
Area of the rooftop = [tex]9x^{2}+6x+1[/tex]
This gives us that a = 9, b = 6 and c = 1
Thus, [tex]x=\frac{-6\pm \sqrt{(6)^{2}-4\times 9\times 1}}{2\times 9}[/tex]
i.e. [tex]x=\frac{-6\pm \sqrt{36-36}}{18}[/tex]
i.e. [tex]x=\frac{-6\pm \sqrt{0}}{18}[/tex]
i.e. [tex]x=\frac{-6}{18}[/tex]
i.e. [tex]x=\frac{-1}{3}[/tex]
So, the area of rooftop = length × width = [tex]9x^{2}+6x+1[/tex] = [tex](x+\frac{1}{3})(x+\frac{1}{3})[/tex].
Thus, we get,
Length of the rooftop = Width of the rooftop = [tex](x+\frac{1}{3})[/tex]
Hence, the quadrilateral is a SQUARE.
Since, the area of the rooftop is given as 361 m².
So, [tex](x+\frac{1}{3})(x+\frac{1}{3})=361[/tex]
i.e. [tex](x+\frac{1}{3})^2=361[/tex]
i.e. [tex]x+\frac{1}{3}=19[/tex]
i.e. [tex]x=19-\frac{1}{3}[/tex]
i.e. [tex]x=\frac{56}{3}[/tex]
So, the length of one side is [tex]x+\frac{1}{3}=\frac{56}{3}+\frac{1}{3}=\frac{57}{3}=19[/tex]
Hence, length of one side of the rooftop is 19 meter.
