A quadratic equation refers to an equation that can be rearranged into standard form [tex]\rm ax^2 + bx + c = 0 \\\\[/tex] .
Quadratic equation formed for the given question will be:
[tex]x^2 + 6x - 187 = 0[/tex]
The answer obtained by solving the equation will be :
[tex]x = 11[/tex] and [tex]y = 17[/tex].
How to form quadratic equation?
Let the two numbers be x and y.
Given:
Product of x and y is 187
Therefore, [tex]xy = 187[/tex] ... (1)
And difference between x and y is 6
Therefore, [tex]x + 6 = y[/tex] ... (2)
On substituting the value of y in equation 1 we get:
[tex]\begin{aligned}x(x+6) &= 187\\x^2 + 6x &= 187\\x^2 + 6x - 187 &= 0\end{aligned}[/tex]
How to solve quadratic equation?
A quadratic equation can be solved by different methods such as factoring, formula method, etc.
Here we will be using formula to solve the equation
[tex]x^2 + 6x - 187 = 0[/tex].
Formula for solving quadratic equation:
[tex]x = \dfrac{-b+\sqrt{b^2-4ac}}{2a}[/tex]
for the given equation, a = 1, b = 6 and c = -187
Putting the values of a, b and c in the formula we get:
[tex]\begin{aligned}x &= \dfrac{-6 \pm \sqrt{6^2 - 4(1)(-187)}}{2}\\\\&= \dfrac{-6\pm \sqrt{36+748} }{2} \\&= \dfrac{-6\pm \sqrt{784}}{2}\\&=\dfrac{-6 \pm 28}{2}\end{aligned}[/tex]
Since, natural numbers cannot be negative, we will solve taking the addition sign.
[tex]\begin{aligned}x &= \dfrac{-6 + 28}{2}\\&= \frac{22}{2} \\&x=11\\\end{aligned}[/tex]
Therefore, as per equation 2
[tex]\begin{aligned}x+6 = y\\11+6 &= y\\17 &= y \end{aligned}[/tex]
Hence, the natural numbers will be 11 and 17.
Learn more about quadratic equations here:
https://brainly.com/question/17177510
