If 3 times a positive number is added to the half of its square the result is 8 find the number
i dont know how to do
plz give solution too
Well, well, well, let's solve this "riddle" together!
So, we have a problem here: 3 times a positive number, let's call it "x", is added to half of its square, and the result is 8. Mathematically, this can be represented as:
3x + (1/2)(x^2) = 8
Now, let's simplify things a bit. To get rid of that pesky fraction, let's multiply everything by 2:
6x + x^2 = 16
Great! We now have a quadratic equation. Let's rearrange it:
x^2 + 6x - 16 = 0
To solve this, we can use the good old quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
Applying it to our equation, we find:
x = (-6 ± √(6^2 - 4(1)(-16))) / (2(1))
x = (-6 ± √(36 + 64)) / 2
x = (-6 ± √100) / 2
x = (-6 ± 10) / 2
Now, before we continue, we need to remember that we're looking for a positive number, right? So, we'll discard the negative value:
x = (-6 + 10) / 2
x = 4 / 2
x = 2
Hooray! The positive number we're looking for is 2. Because when we substitute it back into our original equation, we get:
3(2) + (1/2)(2^2) = 6 + 2 = 8
So, 2 is indeed our answer. Ta-da!
Let's solve the equation step by step to find the number.
Step 1: Let's assume the number to be x.
Step 2: The square of the number is x^2.
Step 3: Half of its square is (1/2)x^2.
Step 4: Three times the number is 3x.
Step 5: The equation can be formed as follows: 3x + (1/2)x^2 = 8.
Step 6: To solve the equation, we need to find the values of x that satisfy the equation.
Step 7: Let's multiply the equation by 2 to remove the fraction: 6x + x^2 = 16.
Step 8: Rearrange the equation: x^2 + 6x - 16 = 0.
Step 9: This is a quadratic equation, which can be factored as follows: (x + 8)(x - 2) = 0.
Step 10: Set each factor equal to zero to find possible solutions: x + 8 = 0 or x - 2 = 0.
Step 11: Solve for x: x = -8 or x = 2.
Step 12: Since the problem states that the number should be positive, we can disregard the solution x = -8.
Step 13: Therefore, the number is x = 2.
To find the number, let's break down the information given in the question.
The problem states that "3 times a positive number" (let's call it x) is added to "the half of its square," which means we need to calculate (1/2) * x^2. The sum of these two amounts is then equal to 8.
So, we can write the equation as follows:
3x + (1/2) * x^2 = 8
To solve this equation, we'll first multiply both sides by 2 to eliminate the fraction:
2(3x + (1/2) * x^2) = 2 * 8
6x + x^2 = 16
Rearranging the equation to make it a quadratic equation:
x^2 + 6x - 16 = 0
To solve this quadratic equation, we can either factor it or use the quadratic formula. If factoring is difficult, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For the equation x^2 + 6x - 16 = 0, a = 1, b = 6, and c = -16.
Substituting the values into the quadratic formula:
x = (-(6) ± √((6)^2 - 4(1)(-16))) / (2(1))
x = (-6 ± √(36 + 64)) / 2
x = (-6 ± √100) / 2
x = (-6 ± 10) / 2
Now, we have two possible solutions for x:
x = (-6 + 10) / 2 = 4/2 = 2
x = (-6 - 10) / 2 = -16/2 = -8
Since the problem states that the number we're looking for is a positive number, the solution is x = 2.
just solve
3x + 1/2 x^2 = 8