Question

Math

Posted 2 months ago

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

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Answer from Sia

Posted 2 months ago

Solution by Steps

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.

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