Question

5x^2-5x+6=0 Is this solution two distinct rational solution, or two distinct irrational solutions, or two complex solutions, or a single rational solution?

Answer 1

Step-by-step explanation

Since we have the function 5x^2-5x+6=0

`\mathrm{For\:a\:quadratic\:equation\:of\:the\:form\:}ax^2+bx+c=0\mathrm{\:the\:solutions\:are\:}`
`x_(1,\:2)=(-b\pm √(b^2-4ac))/(2a)`
`\mathrm{For\:}\quad a=5,\:b=-5,\:c=6`
`x_(1,\:2)=(-\left(-5\right)\pm √(\left(-5\right)^2-4\cdot \:5\cdot \:6))/(2\cdot \:5)`

Computing the powers:

`=√(5^2-4\cdot \:5\cdot \:6)`
`\mathrm{Multiply\:the\:numbers:}\:4\cdot \:5\cdot \:6=120`
`=√(5^2-120)`

Apply imaginary number rule:

`=√(5^2-120)`
`=√(95)i`
`x_(1,\:2)=(-\left(-5\right)\pm √(95)i)/(2\cdot \:5)`
`Separate\:the\:solutions`
`x_1=(-\left(-5\right)+√(95)i)/(2\cdot \:5),\:x_2=(-\left(-5\right)-√(95)i)/(2\cdot \:5)`

Simplifying:

`\mathrm{The\:solutions\:to\:the\:quadratic\:equation\:are:}`
`x=(1)/(2)+i(√(95))/(10),\:x=(1)/(2)-i(√(95))/(10)`

In conclusion, the function have two complex solutions.