Question

Find two positive numbers whose difference is 7 and whose product is 744.​

Answer 1

Answer:

31 and 24

Step-by-step explanation:

Ok so let's just say that are two unknowns are "x" and "y". We can derive the following equation: $x-y=7$, which also means that x > y so x is going to be representing the larger number.

Now using the the fact that the product is 74, we get the following equation: $xy=744$

We can solve for "y" in the second equation, by dividing by "x" on both sides, and then substitute that into the x-y=7 equation

Original Equation:

$xy=744$

Divide both sides by "x"

$y=\frac{744}{x}$

Original Equation:

$x-y=7$

Substitute the 744/x as "y"

$x-\frac{744}{x}=7$

Multiply both sides by x

$x^2-744=7x$

Subtract 7x from both sides

$x^2-7x-744=0$

Now we can just use the quadratic formula: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ to solve for x

In this case, a=1, b=-7, c=-744

$x=\frac{-(-7)\pm\sqrt{(-7)^2-4(1)(-744)}}{2(1)}$

Simplify the discriminant (stuff under radical), the -(-7), and the 2(1)

$x=\frac{7\pm\sqrt{3,025}}{2}$

Simplify the radical:

$x=\frac{7\pm55}{2}$

In the question it states "two positive numbers", so we know that the negative square root, will give us (7-55)/2 and that isn't positive. So we only take the positive square root solution (7+55)/2

$x=\frac{7+55}{2}$

Simplify

$x=\frac{62}{2}\implies x=31$

Now we can use either of the initial equations to solve for "y", but the easiest one is x-y=7

$31-y=7$

Add y to both sides

$31=7+y$

Subtract 7 from both sides

$24=y$

We can make sure this also has a product of 744, by using the second equation:

$(31)(24)=744$

Simplify

$744=744$

So these are the two numbers.