The product of 7 more than a number and 14 more than that same number is 330. What are the possible values of the number? (2 values)
Translation:
(x+7)(x+14) = 330
x^2 + 21x - 232 = 0
(x - 8)(x + 29) = 0
x = 8 or x = -29
Oh, solving math problems, are we? Let me put on my thinking cap, or should I say, my funny hat!
So, we have two values that need to be found, right? Let's call the mysterious number X.
According to the problem, the product of 7 more than X and 14 more than X is 330.
We can write this expression as (X + 7)(X + 14) = 330.
Now, let's put on our detective hats and solve this equation!
Expanding the equation gives us X^2 + 21X + 98 = 330.
Brace yourselves, here comes the funny part...
We want to find two values of X that make this equation true. Well, it looks like we have a quadratic equation on our hands!
Using some mathematical magic, we shift everything to one side and get X^2 + 21X - 232 = 0.
Now, let's call in the funny math circus to find the solutions!
Using the quadratic formula, X = (-b ± √(b^2 - 4ac)) / (2a), we get two possible values for X. And they are...
*drumroll*
X = -29 or X = 8!
Voila! We have our two possible values for the mystery number X.
Let's assume the number is "x".
According to the given information, "The product of 7 more than a number and 14 more than that same number is 330", we can form the equation:
(x + 7) * (x + 14) = 330
Expanding the equation:
x^2 + 14x + 7x + 98 = 330
Combining like terms:
x^2 + 21x + 98 = 330
Rearranging the equation to set it equal to zero:
x^2 + 21x + 98 - 330 = 0
Simplifying:
x^2 + 21x - 232 = 0
Now, we can solve this quadratic equation by factoring or using the quadratic formula.
By factoring, we get:
(x + 29)(x - 8) = 0
Setting each factor to zero and solving for "x":
x + 29 = 0 or x - 8 = 0
x = -29 or x = 8
Therefore, the possible values of the number are -29 and 8.
To find the possible values of the number, let's set up an equation based on the given information.
Let's assume the number is represented by "x."
According to the problem, "7 more than a number" can be expressed as (x + 7). Similarly, "14 more than that same number" can be expressed as (x + 14).
The problem states that the product of these two expressions is 330. So we have the equation:
(x + 7) * (x + 14) = 330
To solve this equation, we can first expand the left side:
x² + 14x + 7x + 98 = 330
Combining like terms, we get:
x² + 21x + 98 = 330
Next, let's simplify the equation by subtracting 330 from both sides:
x² + 21x + 98 - 330 = 0
x² + 21x - 232 = 0
Now, we need to factor this quadratic equation, or we can use the quadratic formula. Let's use the quadratic formula to find the values of "x":
x = (-b ± √(b² - 4ac)) / (2a)
For this equation, a = 1, b = 21, and c = -232.
Substituting these values into the quadratic formula, we get:
x = (-21 ± √(21² - 4(1)(-232))) / (2(1))
Simplifying further:
x = (-21 ± √(441 + 928)) / 2
x = (-21 ± √1369) / 2
x = (-21 ± 37) / 2
This gives us two possible values for "x":
x₁ = (-21 + 37) / 2 = 16 / 2 = 8
x₂ = (-21 - 37) / 2 = -58 / 2 = -29
Therefore, the possible values of the number are 8 and -29.