The Product Of Two Consective Positive Odd Numbers 195, By Constructing A Quadratic Eqation And Solving It, Find The Two Numbers.
first of the numbers ---- x
the next odd number ---- x+2
x(x+2) = 195
x^2 + 2x - 195 = 0
(x+15)(x-13) = 0
carry on
Let's assume that the first odd number is "x" and the second odd number is "x + 2".
We can construct a quadratic equation based on the given information.
The product of two consecutive positive odd numbers, 195, can be expressed as:
x * (x + 2) = 195
Expanding the equation, we have:
x^2 + 2x = 195
Rearranging the equation by subtracting 195 from both sides, we get:
x^2 + 2x - 195 = 0
Now, we need to solve this quadratic equation to find the values of "x" and "x + 2".
To solve the quadratic equation, we can either factorize it or use the quadratic formula.
Factoring this quadratic equation, we find:
(x - 13)(x + 15) = 0
Setting each factor to zero and solving for "x", we get:
x - 13 = 0 --> x = 13
x + 15 = 0 --> x = -15
Since we assumed that the numbers are positive odd numbers, we can discard the negative solution.
Therefore, the first odd number is 13, and the second odd number is 13 + 2 = 15.
So, the two consecutive odd numbers are 13 and 15.
To solve this problem, let's start by assuming the two consecutive odd numbers are x and x+2.
According to the problem, the product of these two numbers is 195. So, we can write the equation as:
x * (x + 2) = 195
Expanding this equation, we get:
x^2 + 2x = 195
Now let's rearrange the equation to form a quadratic equation in standard form:
x^2 + 2x - 195 = 0
To find the values of x that satisfy this equation, we can use the quadratic formula. The quadratic formula states that for a quadratic equation ax^2 + bx + c = 0, the solutions for x can be calculated as:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation x^2 + 2x - 195 = 0, a = 1, b = 2, and c = -195. Substituting these values into the quadratic formula, we get:
x = (-2 ± √(2^2 - 4(1)(-195))) / (2(1))
Simplifying further:
x = (-2 ± √(4 + 780)) / 2
x = (-2 ± √784) / 2
x = (-2 ± 28) / 2
Now, we have two possible solutions for x:
x1 = (-2 + 28) / 2 = 26 / 2 = 13
x2 = (-2 - 28) / 2 = -30 / 2 = -15
Since we need positive odd numbers, we can eliminate the negative number. Therefore, the two consecutive positive odd numbers are 13 and 15, respectively.