When 30 times a number is increased by 32 ,the result is equal to twice the square of the number. Find the number
x = number
When 30 times a number is increased by 32 ,the result is equal to twice the square of the number means:
30 x + 32 = 2 x²
Subtract 2 x² to both sides
- 2 x² + 30 x + 32 = 0
Divide both sides by - 2
x² - 15 x - 16 = 0
The solutions are:
x = - 1
and
x = 16
Check of results:
30 x + 32 = 2 x²
30 ∙ ( - 1 ) + 32 = 2 ∙ ( - 1 )²
-30 + 32 = 2 ∙ 1
2 = 2
30 ∙ 16 + 32 = 2 ∙ 16²
480 + 32 = 2 ∙ 256
512 = 512
512
The number is 16.
Well, well, well, let's solve this equation with a pinch of humor, shall we?
Let's call the number we're looking for "x". According to the problem, when 30 times a number is increased by 32, we get twice the square of the number.
Translated into an equation, that looks like this:
30x + 32 = 2x^2
Now, we have a quadratic equation on our hands! We can solve this bad boy either by factoring, completing the square, or using the quadratic formula.
But since we're in a light-hearted mood, let's use the quadratic formula, shall we?
x = (-b ± √(b² - 4ac)) / (2a)
Plug in our values: a = 2, b = -30, c = -32:
x = (30 ± √((-30)² - 4(2)(-32))) / (2(2))
Now it's just a matter of some arithmetic and algebraic acrobatics! Solving this equation will bring us to the answer we seek. Enjoy the journey!
To find the number, we need to set up an equation based on the given information and then solve for the unknown variable.
Let's start by assigning a variable to the unknown number. Let's call it "x".
According to the information given, "30 times a number is increased by 32" can be written as 30x + 32.
And "twice the square of the number" can be written as 2x^2.
As per the problem, these two expressions are equal. Therefore, we can set up the equation:
30x + 32 = 2x^2
To solve this equation, we'll move all the terms to one side and set it equal to zero:
2x^2 - 30x - 32 = 0
Now, we can solve this quadratic equation using factoring, completing the square, or the quadratic formula. Let's solve it using factoring:
First, we can factor out a common factor of 2:
2(x^2 - 15x - 16) = 0
Next, we need to factor the quadratic expression inside the parentheses:
2(x + 1)(x - 16) = 0
Now we have two factors: (x + 1) and (x - 16). To find the value of x, we need to set each factor equal to zero and solve for x:
x + 1 = 0 or x - 16 = 0
Solving for x in each equation:
x + 1 = 0 => x = -1
x - 16 = 0 => x = 16
So, the possible values for the number are x = -1 or x = 16.