Find two negative numbers such that the sum of their squares is 170 and twice the square of the first minus 3 times the square of the second is 95
a^2+b^2 = 170
2a^2-3b^2 = 95
substituting,
2a^2-3(170-a^2) = 95
5a^2 - 510 = 95
5a^2 = 605
a^2 = 121
a = -11
b = -7
Why did the negative numbers go to therapy? Because they needed some square-ity in their lives!
Let's call the first negative number x and the second negative number y.
According to the problem, we have two conditions:
1) x^2 + y^2 = 170
2) 2x^2 - 3y^2 = 95
Let's solve these equations together, shall we?
First, let's multiply the second equation by (-1) to make the coefficient of y^2 positive:
-2x^2 + 3y^2 = -95
Now, we can add both equations together:
(x^2 + y^2) + (-2x^2 + 3y^2) = 170 + (-95)
Simplifying, we get:
-x^2 + 4y^2 = 75
Let's rearrange this equation:
4y^2 - x^2 = 75
To make things easier, let's set x = y, which gives us:
4y^2 - y^2 = 75
Simplifying further, we get:
3y^2 = 75
Dividing both sides by 3, we find:
y^2 = 25
Taking the square root of both sides, we get:
y = ±5
Since we are looking for negative numbers, let's take y = -5:
Now, let's substitute y = -5 in either of the original equations. Let's use the first equation:
x^2 + (-5)^2 = 170
x^2 + 25 = 170
x^2 = 170 - 25
x^2 = 145
Taking the square root of both sides, we get:
x = ±√145
Since we are looking for negative numbers, let's take x = -√145:
So, the two negative numbers that satisfy the given conditions are:
x = -√145, y = -5
Therefore, the solution is x = -√145 and y = -5.
Let's solve this problem step-by-step.
Step 1: Let's set up the equations using the given information. Let's assume the first negative number is represented by 'x' and the second negative number is represented by 'y.'
Given:
The sum of their squares is 170:
x^2 + y^2 = 170
And,
Twice the square of the first minus 3 times the square of the second is 95:
2x^2 - 3y^2 = 95
Step 2: Rearrange the equations. Let's rewrite the second equation in terms of x:
2x^2 - 3y^2 = 95
2x^2 = 95 + 3y^2
x^2 = (95 + 3y^2)/2
Step 3: Substitute the value of x^2 from the rearranged equation into the first equation:
x^2 + y^2 = 170
(95 + 3y^2)/2 + y^2 = 170
Step 4: Simplify the equation:
95 + 3y^2 + 2y^2 = 340
5y^2 = 340 - 95
5y^2 = 245
Step 5: Divide both sides of the equation by 5:
y^2 = 245/5
y^2 = 49
y = √49
y = -7
Step 6: Substitute the value of y into the first equation to solve for x:
x^2 + (-7)^2 = 170
x^2 + 49 = 170
x^2 = 170 - 49
x^2 = 121
x = √121
x = -11
So, the two negative numbers that satisfy the given conditions are -11 and -7.
To solve this problem, let's start by assigning variables to the two negative numbers. Let the first negative number be represented by x, and the second negative number be represented by y.
We are given two conditions in the problem:
1) The sum of their squares is 170: x^2 + y^2 = 170
2) Twice the square of the first minus 3 times the square of the second is 95: 2x^2 - 3y^2 = 95
We can solve this system of equations simultaneously to find the values of x and y.
First, let's isolate one variable in terms of the other from equation (1). We will solve for y:
y^2 = 170 - x^2 [subtract x^2 from both sides]
y = √(170 - x^2) [take the square root of both sides]
Now, substitute this expression for y into equation (2):
2x^2 - 3(√(170 - x^2))^2 = 95
Simplify the equation:
2x^2 - 3(170 - x^2) = 95
2x^2 - 510 + 3x^2 = 95
5x^2 - 510 = 95
5x^2 = 605
x^2 = 605 / 5
x^2 = 121
x = √121
x = 11
Now substitute the value of x back into equation (1) to solve for y:
11^2 + y^2 = 170
121 + y^2 = 170
y^2 = 170 - 121
y^2 = 49
y = √49
y = 7
So, the two negative numbers that satisfy the given conditions are x = -11 and y = -7.