BIQUADRATE

, or Biquadratic Power, is the squared square, or 4th power of any number or quantity. Thus 16 is the biquadrate or 4th power of 2, or it is the square of 4 which is the 2d power of 2.

Biouadratic Root, of any quantity, is the square root of the square root, or the 4th root of that quantity. So the biquadratic root of 16 is 2, and the biquadratic root of 81 is 3.

Biquadratic Equation, is that which rises to 4 dimensions, or in which the unknown quantity rises to the 4th power; as .

Any biquadratic equation may be conceived to be generated or produced from the continual multiplication of four simple equations, as ; or from that of two quadratic equations, as ; or, lastly, from that of a cubic and a simple equation, as : which was the invention of Harriot. And, on the contrary, a biquadratic equation may be resolved into four simple equations, or into two quadratics, or into a cubic and a simple equation, having all the same roots with it.

The first resolution of a biquadratic equation was given in Cardan's Algebra, chap. 39, being the invention of his pupil and friend Lewis Ferrari, about the year 1540. This is effected by means of a cubic equation, and is indeed a method of depressing the biquadratic equation to a cubic, which Cardan demonstrates, and applies in a great variety of examples. The principle is very general, and consists in completing one side of the equation up to a square, by the help of some multiples or parts of its own terms and an assumed unknown quantity; which it is always easy to do; and then the other side is made to be a square also, by assuming the product of its 1st and 3d terms equal to the square of half the 2d term; for it consists only of three terms, or three different denominations of the original letter; then this equality will determine the value of the assumed quantity by a cubic equation: other circumstances depend on the artist's judgment. But the method will be farther explained by the following examples, extracted from Cardan's book.

Ex. 1. Given , to be resolved. Add 6x² to both sides of the equation, so shall . Assume y, and add to both sides, then is . Make now the , this gives , or ; and hence . From which x may be found by a quadratic equation.

Ex. 2. Given . Before applying Ferrari's method to this example, Cardan resolves it by another way as follows: subtract 1, then is ; divide by x + 1, then is ; and hence .

But to resolve it by Ferrari's rule: Because . theresore ; hence ; and the root is : by means of which x is found by a quadratic equation.

Ex. 3. Given .—Add 240, then ; complete square again, then ; make the last side a sq. by the rule, which gives . Put now , and the last transforms to ; then the value of z found from this, gives the value of y, and hence the value of x, as before. |

Another solution was given of biquadratic equations by Descartes, in the 3d book of his Geometry. In this solution he resolved the given biquadratic equation into two quadratics, by means of a cubic equation, in this manner: First, let the 2d term or 3d power be taken away out of the equation, after which it will stand thus, . Find y in this , and these values of x will be the roots of the given biquadratic equation.

Ex. Let the equ. be , Hence p = - 17, q = - 20, & r = - 6; and the cubic equ. is , the root of which is y² = 16, or y = 4; , the four roots of which are 2 ± √7 and - 2 ± √2.

The celebrated Leonard Euler gave, in the 6th volume of the Petersburgh Ancient Commentaries, for the year 1738, an ingenious and general method of resolving equations of all degrees, by means of the equation of the next lower degree, and among them of the biquadratic equation by means of the cubic; and this last was also given more at large in his treatise of Algebra, translated from the German into French in 1774, in 2 volumes 8vo. The method is this: Let , be the given biquadratic equation, wanting the 2d term. Take with which values of f, g, h, form the cubic equation . find the three roots of this cubic equation, and let them be called p, q, r. Then shall the four roots of the proposed biquadratic be these following, viz,

Ex. Let the eq. be . Here a = 25, b = - 60, and c = 36; theref. f = 25/2, , and h = 225/4. Conseq. the cubic equation will be . The three roots of which are z = 9/4 = p, and z = 4 = q, and z = 25/4 = r; the roots of which are √p = 3/2, √q = 2 or 4/2, √r = 5/2. Hence, as the value of (1/8)b is negative, the four roots are .

Mr. Simpson gave also a general rule for the solution of biquadratic equations, in the 2d edit. of his Algebra, pa. 150, in which the given equation is also resolved by means of a cubic equation, as well as the two former ways; and it is investigated on the principle, that the given equation is equal to the difference between two squares; being indeed a kind of generalization of Ferrari's method.

Thus, he supposes the given equation, viz, ; then from a comparison of the like terms, the values of the assumed letters are found, and the final equation becomes , where . The value of A being found in this cubic equation, from it will be had the values of B and C, which have these general values, viz, . Hence, finally, the root x will be obtained from the assumed equation , in four several values.

Ex. Given the equ. . Here p = -6, q = - 58, r = -114, and s = - 11, whence ; and therefore the cubic equation becomes , the root of which is A = 4. Hence then B or : and the quadratic equation becomes , the four roots of which are .

Mr. Simpson here subjoins an observation which it has since been found is erroneous, viz, that “The value of A, in this equation, will be commensurate and rational (and therefore the easier to be discovered), not only when all the roots of the given equation are commensurate, but when they are irrational and even impossible; as will appear from the examples subjoined.” This is a strange reason for Simpson to give for the proof of a proposition; and it is wonderful that he fell upon no examples that disprove it, as the instances in which it holds true, are very few indeed, in comparison with the number of those in which it fails.

Note. In any biquadratic equation having all its terms, if 3/8 of the square of the coefficient of the 2d term be greater than the product of the coefficients of the 1st and 3d terms, or 3/8 of the square of the coefficient | of the 4th term be greater than the product of the coefficients of the 3d and 5th terms, or 4/9 of the square of the coefficient of the 3d term greater than the product of the coefficients of the 2d and 4th terms; then all the roots of that equation will be real and unequal; but if either of the said parts of those squares be less than either of those products, the equation will have imaginary roots.

For the construction of biquadratic equations, see Construction. See also Descartes's Geometry, with the Commentaries of Schooten and others; Baker's Geometrical Key; Slusius's Mesolabium; l'Hospital's Conic Sections; Wolfius's Elementa Matheseos; &c.

Biouadratic Parabola, a curve of the 3d order, having two infinite legs, and expressed by one of these three equations, viz, a³x = y⁴ , as in fig. 1, , as in fig. 2, , as in fig. 3; where x = AP the absciss, y = PQ the ordinate, b = AB, c = AC, and a = a certain given quantity. Fig1 Fig2 Fig3

But the most general equation of this curve is the sollowing, which belongs to fig. 4, viz, Fig4 ; where x = Ap or AP the absciss, and - y or + y is the ordinate pm or PM, also a, b, c, d, e, are constant quantities; the beginning of the absciss being at any point A in the indefinite line AP.

But if the beginning of the absciss A be where this line intersects the curve, as in fig. 5, then the nature of the curve will be defined by this equation , wherever the point p is taken in the infinite line RS. Fig5 Fig6

When the curve has no serpentine part, as fig. 6, the equation is more simple, being in this case barely . See Curve Lines, and Geometrical Lines.