Question

What is the value of a if a+b=5 and ab=-2​

Answer 1

`a = (5 + √(33))/(2)` or
`a = (5 - √(33))/(2)`

Explanation:

a + b = 5 (Eq. 1)

ab = -2 (Eq. 2)

Solve Eq. 1 for b.

b = 5 - a

Plug in 5 - a for b in Eq. 2.

a(5 - a) = -2

Distribute a.

-a^2 + 5a = -2

Multiply both sides by -1.

a^2 - 5a = 2

Complete the square.

a^2 - 5a + (5/2)^2 = 2 + (5/2)^2

(a - 5/2)^2 = 4/4 + 29/4

a - 5/2 = sqrt(33/4) or a - 5/2 = -sqrt(33/4)

a = 5/2 + sqrt(33)/2 or a = 5/2 - sqrt(33)/2

`a = (5 + √(33))/(2)` or
`a = (5 - √(33))/(2)`

Answer 2

5 ±sqrt( 33)

a= -----------------------------------

2

Explanation:

a+b=5 and ab=-2

Take the second equation and divide by a

ab/a = -2/a

b = -2/a

Substitute this into the first equation

a + -2/a = 5

Multiply each side by a

a( a + -2/a = 5)

Distribute

a^2 -2 = 5a

Subtract 5a from each side

a^2 -5a -2 = 0

Using the quadratic equation

5 ±sqrt( 5^2 - 4*1*(-2))

a= -----------------------------------

2(1)

5 ±sqrt( 25 +8))

a= -----------------------------------

2(1)

5 ±sqrt( 33)

a= -----------------------------------

2