Question

Solve the quadratic equation (√5 – 3)x^2 + 3x + (√5+3) = 0, giving your answers in the form a+b√5, where a and b are constants.​

Answer 1

`x = ((-3)/(4)+(-1)/(4)√(5) ) \:\:and\:\: (3 + √(5))`

Explanation:

For a quadratic equation ax² + bx + c ,

its roots are =
`x = (-b + √(b^2 - 4ac) )/(2a) \:\: and \:\: (-b - √(b^2 - 4ac) )/(2a)`

In the question ,

Quadratic equation = (√5 - 3)x² + 3x + (√5 + 3)

By using quadratic formula ,

`x = \frac{-3 + \sqrt{3^2 - 4*(√(5)-3 )*(√(5)+3 ) } }{2(√(5) -3)} ; \frac{-3 - \sqrt{3^2 - 4 * (√(5)-3 )(√(5) +3)} }{2(√(5) -3)}`

`= \frac{-3 + \sqrt{9 - 4*(√(5)^2-3^2 )} }{2(√(5) -3)} ; \frac{-3 - \sqrt{9 - 4 * (√(5)^2-3^2 )}}{2(√(5) -3)}`

`= (-3 + √(9 - 4*(-4)) )/(2(√(5) -3)) ; (-3 - √(9 - 4 * (-4)))/(2(√(5) -3))`

`= (-3 + √(9 +16) )/(2(√(5) -3)) ; (-3 - √(9 +16))/(2(√(5) -3))`

`= (-3 + 5 )/(2(√(5) -3)) ; (-3 -5)/(2(√(5) -3))`

`= (2 )/(2(√(5) -3)) ; (-8)/(2(√(5) -3))`

`= (1 )/((√(5) -3)) ; (-4)/((√(5) -3))`

Rationalizing the roots ,

`x = (1 * (√(5) + 3) )/((√(5) -3) * (√(5) + 3)) ; (-4 * (√(5) +3))/((√(5) -3)(√(5) +3))`

`= (√(5) + 3)/(-4) ; (-4(√(5) +3))/(-4)`

`= ((-3)/(4)+(-1)/(4)√(5) ) ; (3 + √(5))`