Mister Exam
Factor 4*x^4-4*x^2-7 squared
The solution
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(4 x^{4} - 4 x^{2}\right) - 7$$
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
$$a = 4$$
$$b = -4$$
$$c = -7$$
Then
$$m = - \frac{1}{2}$$
$$n = -8$$
So,
$$4 \left(x^{2} - \frac{1}{2}\right)^{2} - 8$$
$$\left(4 x^{4} - 4 x^{2}\right) - 7$$
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
$$a = 4$$
$$b = -4$$
$$c = -7$$
Then
$$m = - \frac{1}{2}$$
$$n = -8$$
So,
$$4 \left(x^{2} - \frac{1}{2}\right)^{2} - 8$$
Factorization
[src]
/ _____________\ / _____________\ / ___________\ / ___________\ | / 1 ___ | | / 1 ___ | | / 1 ___ | | / 1 ___ | |x + I* / - - + \/ 2 |*|x - I* / - - + \/ 2 |*|x + / - + \/ 2 |*|x - / - + \/ 2 | \ \/ 2 / \ \/ 2 / \ \/ 2 / \ \/ 2 /
$$\left(x - i \sqrt{- \frac{1}{2} + \sqrt{2}}\right) \left(x + i \sqrt{- \frac{1}{2} + \sqrt{2}}\right) \left(x + \sqrt{\frac{1}{2} + \sqrt{2}}\right) \left(x - \sqrt{\frac{1}{2} + \sqrt{2}}\right)$$
(((x + i*sqrt(-1/2 + sqrt(2)))*(x - i*sqrt(-1/2 + sqrt(2))))*(x + sqrt(1/2 + sqrt(2))))*(x - sqrt(1/2 + sqrt(2)))
Combining rational expressions
[src]
2 / 2\ -7 + 4*x *\-1 + x /
$$4 x^{2} \left(x^{2} - 1\right) - 7$$
-7 + 4*x^2*(-1 + x^2)