Twice the square of a positive number increased by three times the number is 14. Find the number.
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2x^2+3x =14
2x^2+3x-14=0
(2x+7)(x-2)=0 (factorize it)
x= -7/2 or x=2 (take the positive number)
To find the number, we can set up an equation based on the given information.
Let's assume the positive number is represented by 'x'.
Twice the square of the number (2x^2) increased by three times the number (3x) is equal to 14.
So, the equation becomes:
2x^2 + 3x = 14
To solve this equation, we rearrange it to bring all the terms to one side:
2x^2 + 3x - 14 = 0
Now we can solve this quadratic equation using factoring, completing the square, or the quadratic formula. In this case, let's use factoring.
We need to find two numbers whose product is equal to the product of the coefficient of x^2 (2) and the constant term (-14), which is -28, and whose sum is equal to the coefficient of x (3).
After some trial and error, we find that the numbers are 7 and -4 (-4 * 7 = -28 and -4 + 7 = 3).
Splitting the middle term 3x as 7x - 4x, we can rewrite the equation as:
2x^2 + 7x - 4x - 14 = 0
Now we can factor by grouping:
x(2x + 7) - 2(2x + 7) = 0
(2x + 7)(x - 2) = 0
Setting each factor equal to zero gives us two possible solutions:
2x + 7 = 0 or x - 2 = 0
Solving each equation yields:
2x = -7 or x = 2
Divide both sides of the first equation by 2:
x = -7/2
However, we were given that the number is positive, so the solution is x = 2.
Thus, the positive number we're looking for is 2.