SOLUTION: Is this correct statement that "Equations {{{ x^2 + x + a = 0 }}} and {{{ x^2 + ax + 1 = 0 }}} have a common real point for exactly one value of a"?

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Question 1184293: Is this correct statement that "Equations +x%5E2+%2B+x+%2B+a+=+0+ and +x%5E2+%2B+ax+%2B+1+=+0+ have a common real point for exactly one value of a"?
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

Is this correct statement that equations
x%5E2%2Bx%2Ba=0+and
x%5E2%2Bax+%2B1=0+
have a common real point for exactly one value of a"?
check if there is a value of a for what discriminants are equal to zero
x%5E2%2Bx%2Ba=0
discriminant:
b%5E2-4ac=0.........in your case a=1, b=1 and c=a
1%5E2-4%2A1%2Aa=0
1+-4a=0
1+=4a+
a=1%2F4

x%5E2%2Bax+%2B1=0
discriminant:
b%5E2-4ac=0.........in your case+a=1, b=a and c=1
a%5E2-4%2A1%2A1=0
a%5E2-4=0
a%5E2=4
a=2 or+a=-2

check if any of a will give us one solution:
a=1%2F4
x%5E2%2Bx%2B1%2F4=0+
x%5E2%2B%281%2F4%29x+%2B1=0
________________________-> no solution exist
a=2
x%5E2%2Bx%2B2=0+
x%5E2%2B2x+%2B1=0
_______________-> no solution exist
a=-2
x%5E2%2Bx-2=0+
x%5E2-2x+%2B1=0
_______________ ->solution exist
parabolas have a common real point for exactly one value of a=-2
so, the statement above is correct