Finding Real roots of a polynomialFinding the real roots of a polynomialPolynomial roots questionShow that the equation $x^4 + rx + s = 0$ has at most two distinct real roots.Polynomial with real rootsThe sum of non real roots of the polynomial equation $x^3+3x^2+3x+3=0$When does $nx^4+4x+3=0$ have real roots?Polynomial with odd number of real rootsCondition on $a$ for $(x^2+x)^2+a(x^2+x)+4=0$Postive and negative real rootsFind the number of distinct real roots of a polynomial
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Finding Real roots of a polynomial
Finding the real roots of a polynomialPolynomial roots questionShow that the equation $x^4 + rx + s = 0$ has at most two distinct real roots.Polynomial with real rootsThe sum of non real roots of the polynomial equation $x^3+3x^2+3x+3=0$When does $nx^4+4x+3=0$ have real roots?Polynomial with odd number of real rootsCondition on $a$ for $(x^2+x)^2+a(x^2+x)+4=0$Postive and negative real rootsFind the number of distinct real roots of a polynomial
$begingroup$
Find all real values of $a$ for which the equation
$(x^2 + a)^2 + a = x$ has four real roots.
Can someone help me with this? I have no idea how to start this.
algebra-precalculus
asked Mar 29 at 2:48
sumisumi
634
$endgroup$
add a comment |
$begingroup$
Find all real values of $a$ for which the equation
$(x^2 + a)^2 + a = x$ has four real roots.
Can someone help me with this? I have no idea how to start this.
algebra-precalculus
asked Mar 29 at 2:48
sumisumi
634
$endgroup$
$begingroup$
It seems as if $a<-0.7$... See here.
$endgroup$
– clathratus
Mar 29 at 2:53
$begingroup$
Is this pre-calculus?
$endgroup$
– Andrei
Mar 29 at 3:15
1
$begingroup$
if it has $4$ real roots, you should be able to write it as a product of two quadratic polynomials with real roots (i.e. positive discriminant)
$endgroup$
– Vasya
Mar 29 at 3:35
add a comment |
$begingroup$
Find all real values of $a$ for which the equation
$(x^2 + a)^2 + a = x$ has four real roots.
Can someone help me with this? I have no idea how to start this.
algebra-precalculus
asked Mar 29 at 2:48
sumisumi
634
$endgroup$
Find all real values of $a$ for which the equation
$(x^2 + a)^2 + a = x$ has four real roots.
Can someone help me with this? I have no idea how to start this.
algebra-precalculus
algebra-precalculus
asked Mar 29 at 2:48
sumisumi
634
asked Mar 29 at 2:48
sumisumi
634
asked Mar 29 at 2:48
sumisumi
634
asked Mar 29 at 2:48
sumisumi
634
asked Mar 29 at 2:48
sumisumi
634
634
$begingroup$
It seems as if $a<-0.7$... See here.
$endgroup$
– clathratus
Mar 29 at 2:53
$begingroup$
Is this pre-calculus?
$endgroup$
– Andrei
Mar 29 at 3:15
1
$begingroup$
if it has $4$ real roots, you should be able to write it as a product of two quadratic polynomials with real roots (i.e. positive discriminant)
$endgroup$
– Vasya
Mar 29 at 3:35
add a comment |
$begingroup$
It seems as if $a<-0.7$... See here.
$endgroup$
– clathratus
Mar 29 at 2:53
$begingroup$
Is this pre-calculus?
$endgroup$
– Andrei
Mar 29 at 3:15
1
$begingroup$
if it has $4$ real roots, you should be able to write it as a product of two quadratic polynomials with real roots (i.e. positive discriminant)
$endgroup$
– Vasya
Mar 29 at 3:35
$begingroup$
It seems as if $a<-0.7$... See here.
$endgroup$
– clathratus
Mar 29 at 2:53
$begingroup$
It seems as if $a<-0.7$... See here.
$endgroup$
– clathratus
Mar 29 at 2:53
$begingroup$
Is this pre-calculus?
$endgroup$
– Andrei
Mar 29 at 3:15
$begingroup$
Is this pre-calculus?
$endgroup$
– Andrei
Mar 29 at 3:15
1
1
$begingroup$
if it has $4$ real roots, you should be able to write it as a product of two quadratic polynomials with real roots (i.e. positive discriminant)
$endgroup$
– Vasya
Mar 29 at 3:35
$begingroup$
if it has $4$ real roots, you should be able to write it as a product of two quadratic polynomials with real roots (i.e. positive discriminant)
$endgroup$
– Vasya
Mar 29 at 3:35
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Hint: $(x^2+a)^2 + a - x = (x^2 + x + a + 1)(x^2 - x + a)$. When do these quadratic factors have two real roots each?
answered Mar 29 at 3:49
Robert IsraelRobert Israel
330k23219473
$endgroup$
add a comment |
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$begingroup$
Hint: $(x^2+a)^2 + a - x = (x^2 + x + a + 1)(x^2 - x + a)$. When do these quadratic factors have two real roots each?
answered Mar 29 at 3:49
Robert IsraelRobert Israel
330k23219473
$endgroup$
add a comment |
$begingroup$
Hint: $(x^2+a)^2 + a - x = (x^2 + x + a + 1)(x^2 - x + a)$. When do these quadratic factors have two real roots each?
answered Mar 29 at 3:49
Robert IsraelRobert Israel
330k23219473
$endgroup$
add a comment |
$begingroup$
Hint: $(x^2+a)^2 + a - x = (x^2 + x + a + 1)(x^2 - x + a)$. When do these quadratic factors have two real roots each?
answered Mar 29 at 3:49
Robert IsraelRobert Israel
330k23219473
$endgroup$
Hint: $(x^2+a)^2 + a - x = (x^2 + x + a + 1)(x^2 - x + a)$. When do these quadratic factors have two real roots each?
answered Mar 29 at 3:49
Robert IsraelRobert Israel
330k23219473
answered Mar 29 at 3:49
Robert IsraelRobert Israel
330k23219473
answered Mar 29 at 3:49
Robert IsraelRobert Israel
330k23219473
answered Mar 29 at 3:49
Robert IsraelRobert Israel
330k23219473
330k23219473
add a comment |
add a comment |
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$begingroup$
It seems as if $a<-0.7$... See here.
$endgroup$
– clathratus
Mar 29 at 2:53
$begingroup$
Is this pre-calculus?
$endgroup$
– Andrei
Mar 29 at 3:15
1
$begingroup$
if it has $4$ real roots, you should be able to write it as a product of two quadratic polynomials with real roots (i.e. positive discriminant)
$endgroup$
– Vasya
Mar 29 at 3:35