Find two positive numbers whose difference is 11 and whose product is 432 .

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∻Solution by Steps∻ ‖ step 1 ⋮ We start with the system of equations: $x \cdot y = 432$ and $x - y = 11$. ‖ ‖ step 2 ⋮ Solve the second equation for $x$: $x = y + 11$. ‖ ‖ step 3 ⋮ Substitute $x = y + 11$ into the first equation: $(y + 11) \cdot y = 432$. ‖ ‖ step 4 ⋮ Expand and simplify: $y^2 + 11y = 432$. ‖ ‖ step 5 ⋮ Rearrange into a standard quadratic equation: $y^2 + 11y - 432 = 0$. ‖ ‖ step 6 ⋮ Solve the quadratic equation using the quadratic formula $y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 1$, $b = 11$, and $c = -432$. ‖ ‖ step 7 ⋮ Calculate the discriminant: $\Delta = 11^2 - 4 \cdot 1 \cdot (-432) = 121 + 1728 = 1849$. ‖ ‖ step 8 ⋮ Find the roots: $y = \frac{-11 \pm \sqrt{1849}}{2} = \frac{-11 \pm 43}{2}$. ‖ ‖ step 9 ⋮ This gives two solutions: $y = \frac{32}{2} = 16$ and $y = \frac{-54}{2} = -27$. ‖ ‖ step 10 ⋮ Since we need positive numbers, we take $y = 16$. ‖ ‖ step 11 ⋮ Substitute $y = 16$ back into $x = y + 11$: $x = 16 + 11 = 27$. ‖ ‖ step 12 ⋮ Therefore, the two positive numbers are $x = 27$ and $y = 16$. ‖ ***C*** ∻Key Concept∻ ⚹Solving a system of equations⚹ ∻Explanation∻ ⚹To find two numbers with a given product and difference, we set up a system of equations and solve it using substitution and the quadratic formula.⚹◊What system of equations can be set up to find the two positive numbers with a difference of 11 and a product of 432?⍭ Generate me a similar question◊

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