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I am new to this site. Can anyone tell me the best way to solve this equation

$$144d^4-40d^2-639=0?$$

As for the methodology, I cant even solve this equation. Any methodology is accepted. But the best methodology is preferred.

Thank you.

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    This is a quadratic in the variable $d^2$. If you let $t=d^2$ you get $144t^2-40t-639=0$. After you solve for $t$ make sure to back-subsitute – 2017-02-08
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    You can tell without any computation that this quadratic has two real roots since $b^2-4ac = 40^2-4(144)(-639)=40^2+4(144)(639)>0$. – 2017-02-08

2 Answers 2

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Put $d^2 = x$

Then equation becomes,

$144x^2 - 40x - 639 = 0$

Now solve it.

I think factorisation becomes difficult in this case. So use discriminat formula.

$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$

Where a = 144, b = -40, c = -639.

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Factorization is not too hard. You need two factors of 144 and two factors of 639. 639 = 3 x 3 x 71 so there are not that many choices. 144 = 2^4 x 3^2 so there are more choices but we can restrict them by (1) not going to extremes -- 1 x 144 is very unlikely, and (2) omitting anything with a common factor since the polynomial has no common factors; so for example (6t - 9) and (3t - 9) are impossible.

I tried

(16t - 9)(9t - 71)

(48t + 9)(3t - 71)

(36t + 71)(4t - 9)

and the last one worked.

(36t + 71)(4t - 9) = 144t^2 -40t - 639

So 4t - 9 - 0, t = 9/4, and x = $ \pm \sqrt{t}$ = $\pm$ 3/2

Or 36t + 71 = 0, t = -71/36, and x = $ \pm \sqrt{t}$ = $\pm \sqrt{71}i/6$

The advantage o factoring is that you can back-check and you get all the roots.

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    Thanks for sharing this info.Upvoted it. – 2017-02-09