Question

Can you help me with my math homework

Answer 1

Q1 - x = -2 and
`x=(1)/(2)`

Q2 -
`x=(9)/(4)` and
`x=(-3)/(2)`

Q3 - Square, Length = 19 m

Explanation:

Question 1:

We have the equation
`2x^(2)+3x=2` i.e.
`2x^(2)+3x-2=0`

The solution of equation
`ax^(2)+bx+c=0` is
`x=\frac{-b\pm \sqrt{b^(2)-4ac}}{2a}`.

So, from the given equation, we get,

a = 2, b = 3 and c = -2.

Thus,
`x=\frac{-3\pm \sqrt{(3)^(2)-4* 2* (-2)}}{2* 2}`

i.e.
`x=(-3\pm √(9+16))/(4)`

i.e.
`x=(-3\pm √(25))/(4)`

i.e.
`x=(-3\pm 5)/(4)`

i.e.
`x=(-3+5)/(4)` and
`x=(-3-5)/(4)`

i.e.
`x=(2)/(4)` and
`x=(-8)/(4)`

i.e.
`x=(1)/(2)` and x= -2

Question 2:

We have the equation
`16x^(2)-12x=54` i.e.
`16x^(2)-12x-54=0`

Again, we have,

a = 16, b = -12 and c = -54

Thus,
`x=\frac{12\pm \sqrt{(-12)^(2)-4* 16* (-54)}}{2* 16}`

i.e.
`x=(12\pm √(144+3456))/(32)`

i.e.
`x=(12\pm √(3600))/(32)`

i.e.
`x=(12\pm 60)/(32)`

i.e.
`x=(12+60)/(32)` and
`x=(12-60)/(32)`

i.e.
`x=(72)/(32)` and
`x=(-48)/(32)`

i.e.
`x=(9)/(4)` and
`x=(-3)/(2)`

Question 3:

We have that,

Area of the rooftop =
`9x^(2)+6x+1`

This gives us that a = 9, b = 6 and c = 1

Thus,
`x=\frac{-6\pm \sqrt{(6)^(2)-4* 9* 1}}{2* 9}`

i.e.
`x=(-6\pm √(36-36))/(18)`

i.e.
`x=(-6\pm √(0))/(18)`

i.e.
`x=(-6)/(18)`

i.e.
`x=(-1)/(3)`

So, the area of rooftop = length × width =
`9x^(2)+6x+1` =
`(x+(1)/(3))(x+(1)/(3))`.

Thus, we get,

Length of the rooftop = Width of the rooftop =
`(x+(1)/(3))`

Hence, the quadrilateral is a SQUARE.

Since, the area of the rooftop is given as 361 m².

So,
`(x+(1)/(3))(x+(1)/(3))=361`

i.e.
`(x+(1)/(3))^2=361`

i.e.
`x+(1)/(3)=19`

i.e.
`x=19-(1)/(3)`

i.e.
`x=(56)/(3)`

So, the length of one side is
`x+(1)/(3)=(56)/(3)+(1)/(3)=(57)/(3)=19`

Hence, length of one side of the rooftop is 19 meter.