Mister Exam
Factor 8*x^4-14*x^2-3 squared
The solution
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4 2 8*x - 14*x - 3
$$\left(8 x^{4} - 14 x^{2}\right) - 3$$
8*x^4 - 14*x^2 - 3
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(8 x^{4} - 14 x^{2}\right) - 3$$
To do this, let's use the formula
$$a x^{4} + b x^{2} + c = a \left(m + x^{2}\right)^{2} + n$$
where
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
In this case
$$a = 8$$
$$b = -14$$
$$c = -3$$
Then
$$m = - \frac{7}{8}$$
$$n = - \frac{73}{8}$$
So,
$$8 \left(x^{2} - \frac{7}{8}\right)^{2} - \frac{73}{8}$$
$$\left(8 x^{4} - 14 x^{2}\right) - 3$$
To do this, let's use the formula
$$a x^{4} + b x^{2} + c = a \left(m + x^{2}\right)^{2} + n$$
where
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
In this case
$$a = 8$$
$$b = -14$$
$$c = -3$$
Then
$$m = - \frac{7}{8}$$
$$n = - \frac{73}{8}$$
So,
$$8 \left(x^{2} - \frac{7}{8}\right)^{2} - \frac{73}{8}$$
Factorization
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/ ______________\ / ______________\ / ____________\ / ____________\ | / ____ | | / ____ | | / ____ | | / ____ | | / 7 \/ 73 | | / 7 \/ 73 | | / 7 \/ 73 | | / 7 \/ 73 | |x + I* / - - + ------ |*|x - I* / - - + ------ |*|x + / - + ------ |*|x - / - + ------ | \ \/ 8 8 / \ \/ 8 8 / \ \/ 8 8 / \ \/ 8 8 /
$$\left(x - i \sqrt{- \frac{7}{8} + \frac{\sqrt{73}}{8}}\right) \left(x + i \sqrt{- \frac{7}{8} + \frac{\sqrt{73}}{8}}\right) \left(x + \sqrt{\frac{7}{8} + \frac{\sqrt{73}}{8}}\right) \left(x - \sqrt{\frac{7}{8} + \frac{\sqrt{73}}{8}}\right)$$
(((x + i*sqrt(-7/8 + sqrt(73)/8))*(x - i*sqrt(-7/8 + sqrt(73)/8)))*(x + sqrt(7/8 + sqrt(73)/8)))*(x - sqrt(7/8 + sqrt(73)/8))
Combining rational expressions
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2 / 2\ -3 + 2*x *\-7 + 4*x /
$$2 x^{2} \left(4 x^{2} - 7\right) - 3$$
-3 + 2*x^2*(-7 + 4*x^2)