Question

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Solve for x using the quadratic formula.
x2 - 2x+5 = 0​

Answer 1

In the set of real numbers, the equation has no solution.

In the set of complex numbers the solutions are:

1 - 2i and 1 + 2i

Explanation:

`\text{The quadratic formula of a quadratic equation}\ ax^2+bx+c=0:\\\\\text{if}\ b^2-4ac<0,\ \text{then an equation has no real solution}\\\\\text{if}\ b^2-4ac=0,\ \text{then an equation has one solution}\ x=(-b)/(2a)\\\\\text{if}\ b^2-4ac>0,\ \text{then an equation has two solutions:}\ x=(-b\pm√(b^2-4ac))/(2a)`

`\text{We have the equation:}\ x^2-2x+5=0\\\\a=1,\ b=-2,\ c=5\\\\\text{Substitute:}\\\\b^2-4ac=(-2)^2-4(1)(5)=4-20=-16<0\\\\\bold{NO\ REAL\ SOLUTION}`

`\text{If you want the complex solution, then:}\\\\i=√(-1)\\\\\text{Therefore}\ √(-16)=√((16)(-1))=√(16)\cdot√(-1)=4i\\\\x=(-(-2)\pm4i)/(2(1))=(2\pm4i)/(2)=1\pm2i`