If the difference of two numbers is 9 and their product is 19, what is the sum of their squares?

Question

We know the answer (from the answer sheet), but can't understand how to get to it. It's driving us nuts!
thanks

Answer 1

(a - b)^2 = a^2 + b^2 - 2 a b

in other words

(diff)^2 = (sum of the squares) - 2 * (product)

9^2 = (sum of squares) - 2 * 19

81 = (sum of squares) - 38

sum of squares = 119

============================================

It is actually possible to calculate the two numbers, but that takes more work.

Since a - b = 9, we have a = b + 9.
Therefore

19 = a *b = b * (b + 9)

so

b^2 + 9 b - 19 = 0

Complete the square (or use quad. formula):

(b + 9/2)^2 - 19 = (9/2)^2
(b + 9/2)^2 = 157/4
b = (-9 +/- sqrt 157) / 2

b = 1.764982043071 and a = 10.764982043071
or
b = -10.764982043071 and a = -1.764982043071

you can check that the product is 19 and the sum of squares 119...

Answer 2

By now you must be up to here with the quadratic formula. How about the most direct way, brute force.
Construct 2 numbers with a difference of 9. Like 1 & 10.
Their product is 10. Too low, so increase the pair to , say
2 & 11. Product is 22. Too high. But you've bracketed
19. So pick a pair 1/2 way between 1,10 and 2,11, say
1.5 & 10.5 etc. It's called a binary search. Sorry. You knew that. I did that and came up with
1.76498211 & 10.76498211 which gives a product of
19.00000084. Not bad. The sum of their squares is
119.0000016.

Answer 3

I'll designate the larger number b and the smaller number a

You are given the following two equations...
--- b=a+9
--- ab=19

Substituting a+9 for b in the second equation you get...
--- a(a+9)=19
--- a^2+9a-19=0

Now all you need to do is solve for a using the quadratic formula...

a = [-9 +- sqrt(9^2 - (-19)*4)]2
a = [-9 +- sqrt(81 + 76)]2
= -4.5 +- 0.5*sqrt(157)

b is then easy to figure out as we just add 9 to it...

b = 4.5 +- 0.5*sqrt(157)

Note that there are two solutions to the problem. The "second" a will just wind up being the negative of the "first" b and vice versa.

Answer 4

You need to set up 2 equations:

(1) A - B = 9
(2) A x B = 19

Solve eq(2) for B --> B = 19/A and substitute it into eq(1)

A - 19/A = 9

Multiply through by A to remove the denominator

A^2 - 19 = 9A

bring the 9A to the left by subtracting it from both sides of the = sign.

A^2 - 9A - 19 = 0

Using the quadratic equation, we find that there are 2 possible answers for A

A = 10.765 or A = -1.765

Plug these into either of the two origional equations and solve for B.

Now, simply calculate A^2 + B^2 = (you do the math)

Answer 5

19

Answer 6

Since 19 is a prime number the difference between two numbers cant be 9.(only products are 1 and 19)
So, this question is wrong

Answer 7

Let the two numbers be x and y
x - y = 9
x * y = 19

x = 9 + y (subsitute into second equation)

(9 + y) * y = 19
y^2 + 9y = 19
y^2 + 9y -19 = 0

Use quadratic formula:
y = [-9 +- sqrt(81 + 4*19)] / 2
y = 1.765
x = 10.765

x^2 + y^2 = 119

or y = -10.765 and x = -1.765
and x^2 + y^2 = 119

Answer 8

x - y = 9 difference
x * y = 19 product

x - 9 = y
by substitution:
(x - 9) * x = 19
x^2 - 9x = 19
x^2 - 9x - 19 = 0

using the quadratic formula:
x = (-b +/- SQRT(b^2 - 4ac))/2a where a = 1, b = -9, c = -19

there are two possible values for x and y

x = (9 + SQRT(157))/2, y = ((9 + SQRT(157))/2) - 9
and
x = (9 - SQRT(157)/2, y = ((9 - SQRT(157))/2) - 9

Answer 9

What Jamie mentioned. in simple terms needed to point that a amazing variety must be extra advantageous than a million. even nevertheless a million has "no divisors different than itself and a million," it fairly is not top. extra suited to declare a amazing variety is a staggering integer that has precisely 2 diverse divisors, itself and a million. top numbers have rather some residences that a million does not have, so because it rather is why a million is excluded from the primes with the help of mathematicians.

Answer 10

x - y = 9
xy = 19

x = y + 9

(y + 9)y = 19
y^2 + 9y = 19
y^2 + 9y - 19 = 0

y = (-b ± sqrt(b^2 - 4ac))/(2a)

y = (-9 ± sqrt(81 - 4(1)(-19)))/2
y = (-9 ± sqrt(81 + 76))/2
y = (-9 ± sqrt(157))/2

x = (1/2)(-9 ± sqrt(157)) + 9
x = (1/2)(9 ± sqrt(157))

((1/2)(9 ± sqrt(157)))^2 + ((1/2)(-9 ± sqrt(157)))^2

(1/4)(9 + sqrt(157))(9 + sqrt(157)) + (1/4)(-9 + sqrt(157))(-9 + sqrt(157))

(1/4)(81 + 9sqrt(157) + 9sqrt(157) + 157) + (1/4)(81 - 9sqrt(157) - sqrt(157) + 157)

(1/4)(238 + 18sqrt(157)) + (1/4)(238 - 18sqrt(157))

(1/4)((238 + 18sqrt(157)) + (238 - 18sqrt(157))

(1/4)(238 + 18sqrt(157) + 238 - 18sqrt(157))

(1/4)(476)

119

ANS : 119