Solve $(x^2 + 5)^2 - 15(x^2 + 5) + 54 = 0$

I got the square root of 14 and 11 but the answer book states that these answers are wrong. Can someone help me? I used this formula to find the individual roots

$x = -\frac{p}{2} \pm \sqrt{(\frac{p}{2})^2 - q}$

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Try this: define a new variable z=x^2+5 and solve z^2-15z+54 using the quadratic formula. Then solve for x in terms of z. – bill s May 15 '13 at 10:24
OP please mark one of these answers right - it's good SE practice and people will answer you more often! – Soham Chowdhury May 17 '13 at 12:44

Let $a = (x^2 + 5)$. Then $$(a-9)(a-6)=0$$

$$\implies a = 9\; or \;a = 6$$ $$\implies x^2+5 = 9 \; or \;6$$ $$\implies x = \pm\sqrt4\; or \pm\sqrt1$$ $$\implies x = \pm2\; or \pm1$$

This kind of equation is called a biquadratic equation. Cheers!

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Using middle term factor, $$(x^2+5)^2-(9+6)(x^2+5)+6\cdot9=0$$

$$\implies (x^2+5)(x^2+5-9)-6(x^2+5-9)=0$$

$$\implies (x^2-4)(x^2-1)=0$$

$\implies x^2-1=0$ or $x^2-4=0$

Alternatively, using quadratic equation formula for $x^2+5=\frac{15\pm\sqrt{15^2-4\cdot1\cdot 54}}{2\cdot1}=\frac{15\pm 3}2=9$ or $6$

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I really know this is offtopic, but it's nice to see a Bengali here :) – Soham Chowdhury May 15 '13 at 11:37
@SohamChowdhury, where are you actually from? – lab bhattacharjee May 15 '13 at 11:40
I'm currently in 9th grade (in Bosco, if you know it) and I'm sitting in an apartment in Dum Dum :) – Soham Chowdhury May 15 '13 at 11:52
This answer has one-way implications. Presumably they should be two-way implications, i.e., logical equivalences; otherwise, you could be obtaining extraneous roots or numbers that are not roots at all. – murray May 15 '13 at 14:01
@murray, not sure how the above method can introduce extraneous roots? – lab bhattacharjee May 15 '13 at 14:38

Given:

$(x^2 + 5)^2 - 15(x^2 + 5) + 54 = 0$

Let a = $(x^2+5)$ which gives us a simple quadratic trinomial of the form $ax^2+bx+c=0.$

$a^2-15a+54=0$

Factoring this we get:

$(a-9)(a-6)=0$

$a-9=0$

$a-6=0$

$\implies a=9$ or $a=6$

$x^2+5=9$

$x^2+5=6$

$\implies x = \pm2$ or $\pm 1$

You can check that these solutions are correct by simply plugging values back in to see if you get $0$.

$a^2-15a+54=0 \implies$ $(9)^2 - 15(9) + 54 = 0 \implies 81-135+54=0$

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