can someone help me out with these problems with explanations please

1.) if 2x^2 + p = 3x

find a value of P so that there are two rational solutions

ive already done the problem however i got anything less than 9/8 (using the discriminant and inequalities) then i checked the book for possible values and it showed up as 5/8? how does that work?

2.) The height of a firework is modelled by the equation h(t) = -at^2 + 5t where h(t) is the height of the firework and a is greater than 0

a.) factorise the function and thus determine when, in terms of a, it would land on the ground (h(t) = 0) if the firework was to follow this path completely

b.) Making the equation equal to d, solve for t in terms of a and d

3.) Can someone define the differences between an equation and an expression?

4.) A rectangle has length l and width w.

a.) write an expression for the perimeter and area in terms of l and w

b.) if the perimeter is 100m, and the area is 75m^2, change the expressions from part (a) into equations

c.) Solve the equations to find the dimensions of the rectangle correc to three decimal places

i once again attempted this and solved it wiht the values 49.24 and 0.7625 using the quadratic formula. I thought that the formula only showed the width values, how come its showing both the L and W?

Thanks guys,
Much Appreciated.

2. Originally Posted by andrew2322
can someone help me out with these problems with explanations please

1.) if 2x^2 + p = 3x

find a value of P so that there are two rational solutions

ive already done the problem however i got anything less than 9/8 (using the discriminant and inequalities) then i checked the book for possible values and it showed up as 5/8? how does that work?
p < 9/8 is correct.

unless there were extra restrictions on p

Originally Posted by andrew2322
2.) The height of a firework is modelled by the equation h(t) = -at^2 + 5t where h(t) is the height of the firework and a is greater than 0

a.) factorise the function and thus determine when, in terms of a, it would land on the ground (h(t) = 0) if the firework was to follow this path completely

b.) Making the equation equal to d, solve for t in terms of a and d
what have you tried here?

3.) Can someone define the differences between an equation and an expression?
an equation has "=" in it. that is, we call it an equation if we have two (or more) mathematical expressions related by an equal sign. expressions are numbers and variables with some mathematical operation on them, that in some way makes sense.

3. hey

hey i was really stumped on that one, couldnt do any bit of it.

4. Originally Posted by andrew2322
hey i was really stumped on that one, couldnt do any bit of it.
how would you factorize -at^2 + 5t? you'd pull out the common term(s) right? here, that's t. so what would we get?

5. hey

in reply to ur private message, doesnt 2 RATIONAL solutions make it mean that they have to be perfect squares?

6. factorising

-at^2 +5t

t(-at +5)

i could all that, but it didnt make any sense to me so i stopped, especially after i saw that in terms of a and d bit, considering there isnt a D variable.

7. Originally Posted by andrew2322
4.) A rectangle has length l and width w.

a.) write an expression for the perimeter and area in terms of l and w
expression for perimeter: 2l + 2w

expression for area: lw

b.) if the perimeter is 100m, and the area is 75m^2, change the expressions from part (a) into equations
we have 2l + 2w = 100

and lw = 75

c.) Solve the equations to find the dimensions of the rectangle correc to three decimal places
you have simultaneous equations here. solve for either l or w from either equation and plug that expression into the other equation. you can then solve it

once you solve for either the length or width, you can plug in whichever one you found into the lw = 75 equation and solve for the other

8. Originally Posted by andrew2322
in reply to ur private message, doesnt 2 RATIONAL solutions make it mean that they have to be perfect squares?
ah, yes. i didn't see the "rational" part. we want $\displaystyle b^2 - 4ac$ to be a perfect square. that is, $\displaystyle 9 - 8p$ must be a perfect square. we want a value of p, so one way to find one is to make p such that it cancels the 8. so p equals (something)/8. so we have 9 minus an integer and we want this to be a perfect square. the nonnegative perfect squares that are less than or equal to 9 are 0, 1, 4, 9. so we want 9 - 8p to be equal to 0, 1, 4, or 9. the book seems to pick 9 - 8p = 4

Originally Posted by andrew2322
-at^2 +5t

t(-at +5)

i could all that, but it didnt make any sense to me so i stopped, especially after i saw that in terms of a and d bit, considering there isnt a D variable.
they are saying, apparently, that they want you to make h(t) = d

so your equation is: d = -at^2 + 5t, and now solve for t. that is part (b)

part (a) wants you to solve -at^2 + 5t = 0 for a