The roots of 2x^2-3x=4 are a and b. Find the simplest quadratic equation which has roots 1/a and 1/b
1 answer:
Find the roots
solve
we use hmm, completing the suare
2(x²-1.5x)=4
divide both sides by 2
x²-1.5x=2
take 1/2 of linear coeiftn and square it
-1.5/2=-0.75, (-0.75)²=0.5625
add that to both sides
x²-1.5x+0.5625=2+0.5625
factor perfect squaer trinomial
(x-0.75)²=2.5625
square root both sides, remember to take positive and negative square roots
x-0.75=+/-√2.5625
add 0.75 to both sides
x=0.75+/-√2.5625
the roots are x=0.75+√2.5625 and x=0.75-√2.5625
1/a and 1/b
1/(0.75+√2.5625) and 1/(0.75-√2.5625)
if the roots of a quadratic equation are r1 and r2 then it factors to
(x-r1)(x-r2)
so then we can factor our equation to be
if we were to try and expand it, we would get
x²+0.75x-0.5
that's the simpliest equation with roots 1/a and 1/b where a and b are he roots of 2x²-3x=4
x²+0.75x-0.5 is answer
solve
we use hmm, completing the suare
2(x²-1.5x)=4
divide both sides by 2
x²-1.5x=2
take 1/2 of linear coeiftn and square it
-1.5/2=-0.75, (-0.75)²=0.5625
add that to both sides
x²-1.5x+0.5625=2+0.5625
factor perfect squaer trinomial
(x-0.75)²=2.5625
square root both sides, remember to take positive and negative square roots
x-0.75=+/-√2.5625
add 0.75 to both sides
x=0.75+/-√2.5625
the roots are x=0.75+√2.5625 and x=0.75-√2.5625
1/a and 1/b
1/(0.75+√2.5625) and 1/(0.75-√2.5625)
if the roots of a quadratic equation are r1 and r2 then it factors to
(x-r1)(x-r2)
so then we can factor our equation to be
if we were to try and expand it, we would get
x²+0.75x-0.5
that's the simpliest equation with roots 1/a and 1/b where a and b are he roots of 2x²-3x=4
x²+0.75x-0.5 is answer
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The equation gives the height of the ball. That is, h is the height of the ball. t is the time. Since we are looking for the time at which the height is 8 (h=8), we need to set the equation equal to 8 and solve for t. We do this as follows:
This is a quadratic equation and as it is set equal to 0 we can solve it using the quadratic formula. That formula is:
You might recall seeing this as "x=..." but since our equation is in terms of t we use "t-=..."
In order to use the formula we need to identify a, b and c.
a = the coefficient (number in front of)
= 16.
b = the coefficient of t = -60
c = the constant (the number that is by itself) = 7
Substituting these into the quadratic formula gives us:
As we have "plus minus" (this is usually written in symbols with a plus sign over a minus sign) we split the equation in two and obtain:
and
So the height is 8 feet at t = 3.63 and t=.12
It should make sense that there are two times. The ball goes up, reaches it's highest height and then comes back down. As such the height will be 8 at some point on the way up and also at some point on the way down.
This is a quadratic equation and as it is set equal to 0 we can solve it using the quadratic formula. That formula is:
You might recall seeing this as "x=..." but since our equation is in terms of t we use "t-=..."
In order to use the formula we need to identify a, b and c.
a = the coefficient (number in front of)
b = the coefficient of t = -60
c = the constant (the number that is by itself) = 7
Substituting these into the quadratic formula gives us:
As we have "plus minus" (this is usually written in symbols with a plus sign over a minus sign) we split the equation in two and obtain:
and
So the height is 8 feet at t = 3.63 and t=.12
It should make sense that there are two times. The ball goes up, reaches it's highest height and then comes back down. As such the height will be 8 at some point on the way up and also at some point on the way down.