Write a quadratic equation to find two consecutive odd natural numbers whose product is 63. Then find the numbers.

easy way -- what are two factors of 63 that are close together?

algebraic way
x(x+2) = 63
x^2 + 2x - 63 = 0
Now we're back to the original question. Factors of 63 that differ by 2.

another way
63 = 8^2-1 = (8-1)(8+1)

To find two consecutive odd natural numbers whose product is 63, we can start by assuming that the first odd number is "x". The next consecutive odd number will be "x + 2" since consecutive odd numbers have a difference of 2.

Now, let's set up the quadratic equation using the given information:

x * (x + 2) = 63

Expanding the equation:

x^2 + 2x = 63

Rearranging the equation to the standard form:

x^2 + 2x - 63 = 0

Now we can solve this quadratic equation to find the values of "x".

Using factoring, we can factorize the equation as:

(x + 9)(x - 7) = 0

Setting each factor to zero:

x + 9 = 0 or x - 7 = 0

Solving for "x":

x = -9 or x = 7

Since we need to find natural numbers, we can discard the negative value. Therefore, the first odd number is 7.

The next consecutive odd number will be:

7 + 2 = 9

So, the two consecutive odd natural numbers whose product is 63 are 7 and 9.

To find the quadratic equation, let's assume the two consecutive odd numbers are x and x + 2.

Their product is 63, so we can set up the equation:

x(x + 2) = 63

Expanding the equation, we get:

x^2 + 2x = 63

Rearranging the equation, we have:

x^2 + 2x - 63 = 0

Now, we can solve this quadratic equation to find the values of x.

To solve the quadratic equation, we can either factor it, complete the square, or use the quadratic formula. In this case, it's not easy to factor the equation, so let's use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

For the equation x^2 + 2x - 63 = 0, a = 1, b = 2, and c = -63. Plugging these values into the quadratic formula, we get:

x = (-2 ± √(2^2 - 4(1)(-63))) / (2(1))

Simplifying further:

x = (-2 ± √(4 + 252)) / 2

x = (-2 ± √256) / 2

x = (-2 ± 16) / 2

Now we have two possible solutions:

1. x = (-2 + 16) / 2 = 14 / 2 = 7
2. x = (-2 - 16) / 2 = -18 / 2 = -9

Since we are looking for odd natural numbers, we can discard the negative value -9, as it is not a natural number.

Therefore, the first odd number (x) is 7, and the next odd number (x + 2) is 7 + 2 = 9.

So, the two consecutive odd numbers whose product is 63 are 7 and 9.