Question

The sum of two numbers is 25 and their product is 144. Find the numbers

Answer 1

Hello!

Answer:

`\Large \boxed{\sf 16 ~and~ 9}`

Explanation:

→ We want to find the numbers whose sum is 25 and their product if 144.

→ Let x and y be the two numbers.

→ So we have these two equations:

`\sf x +y = 25`

`\sf x* y = 144`

→ Let's solve x in the first equation:

◼ Subtract y from both sides:

`\sf x +y-y = 25-y`

◼ Simplify both sides:

`\sf x = 25-y`

→ Let's replace x by 25 - y in the seconde equation:

`\sf (25 - y)* y = 144`

→ Let's solve this equation to find x and y:

◼ Simplify the left side:

`\sf 25* y - y* y = 144`

`\sf 25y - y^2 = 144`

◼ Put the equation to 0:

`\sf - y^2 +25y -144=0`

→ It's a quadratic equation because it's on the form ay² + by + c = 0.

→ To solve a quadratic equation, there is the quadratic formula:

`\sf y =(-b\pm√(b^2 - 4ac) )/(2a)`

In our equation:

`\sf a =-1\\b = 25\\c = -144`

→ Let's apply the quadratic formula:

`\sf y =(-25\pm√(25^2 - 4(-1)(-144)) )/(2(-1))`

→ Simplify the equation:

`\sf y =(-25\pm√(49) )/(-2)`

`\sf y =(-25\pm7 )/(-2)`

→ Find the two solutions:

`\sf y_1 =(-25+7 )/(-2) = (-18 )/(-2) = 9`

`\sf y_2 =(-25-7 )/(-2) = (-32 )/(-2) = 16`

→ Now we know that y = 9 or 16.

We know that
`\sf 16 + 9 = 25`, and
`\sf 16 * 9 = 144`, so there is no need to calculate x since we already have the 2 numbers.

So the two numbers are 9 and 16.

Conclusion:

The two numbers whose sum is 25 and their product if 144 are 9 and 16.

Answer 2

let x and y be the numbers

x+y=25

xy=144

per substitution: x=25-y

we replace x in the other equation

→(25-y)y=144

25y-y²-144=0

-y²+25y-144=0

values of the coefficients: a=−1; b=25; c=−144

we calculate delta: b²-4ac = (25)²-4(-1*-144)=49

√Δ=7

Δ is strictly positive, the equation −x2+25x−144=0 admits two solutions

(-b-√Δ)/2a = (-25-7)/-2=16

(-b+√Δ)/2a =(-25+7)/-2=9

numbers: 16 and 9