one number is 5 more than half a second number, and the difference between the squares is 17. find the numbers
one number --- 2x , (avoiding fractions)
the other ----- x + 5
(2x)^2- (x+5)^2 = 17
4x^2 - x^2 - 10x -25 - 17 = 0
3x^2 - 10x - 42 = 0
x = (10 ± √604)/6
= (5 ± √151)/3
2x = ....
x+5 = ...
it works, I checked on my calculator
Let's represent the two numbers as x and y.
According to the given information, one number is 5 more than half of the second number, so we can write the first equation as:
x = (1/2)y + 5
The difference between the squares of the two numbers is 17, so we can write the second equation as:
x^2 - y^2 = 17
To solve this system of equations, we can substitute the value of x from the first equation into the second equation:
((1/2)y + 5)^2 - y^2 = 17
Expanding and simplifying this equation, we get:
(1/4)y^2 + 5y + 25 - y^2 = 17
Combine like terms:
(1/4)y^2 - y^2 + 5y + 25 = 17
Multiplying both sides of the equation by 4 to eliminate the fraction:
y^2 - 4y + 100 = 68
Rearranging the equation:
y^2 - 4y + 32 = 0
Now we can solve this quadratic equation for y by factoring, completing the square, or using the quadratic formula. However, this equation does not factor easily, so let's use the quadratic formula:
y = (-(-4) ± √((-4)^2 - 4(1)(32))) / (2(1))
Simplifying:
y = (4 ± √(16 - 128)) / 2
y = (4 ± √(-112)) / 2
Since the square root of a negative number is not a real number, there are no real solutions for y. Therefore, there are no real solutions for x either.
In conclusion, there are no real numbers that satisfy the given conditions.
To solve this problem, let's assign variables to the two numbers. Let's call the first number x and the second number y.
According to the problem, you can set up two equations:
1) "One number is 5 more than half a second number":
x = (1/2)y + 5
2) "The difference between the squares is 17":
x^2 - y^2 = 17
Now we have a system of two equations with two variables. We can solve this system using substitution or elimination.
Let's use substitution to solve:
Substitute the value of x from the first equation into the second equation:
((1/2)y + 5)^2 - y^2 = 17
Square both sides:
(1/4)y^2 + 2.5y + 5^2 - y^2 = 17
Simplify:
(1/4)y^2 - y^2 + 2.5y + 25 - 17 = 0
Combine like terms:
(-3/4)y^2 + 2.5y + 8 = 0
Multiply both sides by -4 to eliminate the fraction:
3y^2 - 10y - 32 = 0
Now we have a quadratic equation that we can solve using factoring, completing the square, or quadratic formula. Let's use factoring:
(3y + 4)(y - 8) = 0
Set each factor equal to zero and solve for y:
3y + 4 = 0 --> 3y = -4 --> y = -4/3
y - 8 = 0 --> y = 8
We have two possible values for y: y = -4/3 and y = 8.
Now substitute the values of y back into the first equation to find the corresponding values of x:
When y = -4/3:
x = (1/2)(-4/3) + 5 --> x = -2/3 + 5 --> x = 14/3
When y = 8:
x = (1/2)(8) + 5 --> x = 4 + 5 --> x = 9
So the solutions to the problem are x = 14/3 and y = -4/3, or x = 9 and y = 8.