QUADRAT
, called also Geometrical Square, and Line of Shadows: it is often an additional member on the face of Gunter's and Sutton's quadrants; and is chiefly useful in taking heights or depths. See my Mensuration, the chap. on Altimetry and Longimetry, or Heights and Distances.
Quadrat, in Astrology, is the same as quartile, being an aspect of the heavenly bodies when they are distant from each other a quadrant, or 90°, or 3 signs, and is thus marked □.
QUADRATIC Equations, in Algebra, are those in which the unknown quantity is of two dimensions, or raised to the 2d power. See Equation.
Quadratic equations are either simple, or affected, that is compound.
A Simple Quadratic equation, is that which contains the 2d power only of the unknown quantity, without any other power of it: as x^{2} = 25, or y^{2} = ab. And in this case, the value of the unknown quantity is found by barely extracting the square root on both sides of the equation: so in the equations above, it will be x = ± 5, and y = ± √ab; where the sign of the root of the known quantity is to be taken either plus or minus, for either of these may be considered as the sign of the value of the root x, since either of these, when squared, make the same square, , and also; and hence the root of every quadratic or square, has two values.
Compound or Affected Quadratics, are those which contain both the 1st and 2d powers of the unknown quantity; as , or , where n may be of any value, and then x^{n} is to be considered as the root or unknown quantity.
Affected quadratics are usually distinguished into three forms, according to the signs of the terms of the equation: Thus, . But the method of extracting the root, or finding the value of the unknown quantity x, is the same in all of them. And that method is usually performed by what is called completing the square, which is done by taking half the coefficient of the 2d term or single power of the unknown quantity, then squaring it, and adding that square to both sides of the equation, which makes the unknown side a complete square. Thus, in the equation , the coefficient of the 2d term being a, its half is (1/2)a, the square of which is (1/4)a^{2}, and this added to both sides of the equation, it becomes , the former side of which is now a complete square. The square being thus completed, its root is next to be extracted; in order to which, it is to be observed that the root on the unknown side consists of two terms, the one of which is always x the square root of the first term of the equation, and the other part is (1/2)a or half the coefficient of the 2d term: thus then the root of x^{2} + ax + (1/4)a^{2} the first side of the completed equation being x + (1/2)a, and the root of the other side (1/4)a^{2} + b being ± √(a^{2} + b), it follows that , and hence, by transposing (1/2)a, it is , the two values of x, or roots of the given equation . And thus is found the root, or value of x, in the three forms of equations above mentioned: thus, . | Where it is observable that, because of the double sign ±, every form has two roots: in the 1st and 2d forms those roots are the one positive and the other negative, the positive root being the less of the two in the 1st form, but the greater in the 2d form; and in the 3d form the roots are both positive. Again, the two roots of the 1st and 2d forms, are always both of them real; but in the 3d form, the two roots are either both real or both imaginary, viz, both real when (1/4)a^{2} is greater than b, or both imaginary when (1/4)a^{2} is less than b, because in this case (1/4)a^{2} - b will be a negative quantity, the root of which is impossible, or an imaginary quantity.
Example of the 1st form, let . Here then a = 6, and b = 7; then .
Example of the 2d form, let . Here also a = 6, and b = 7; then ; the same two roots as before, with the signs changed.
Example of the 3d form, let . Here again a = 6, and b = 7; then , the two roots both real.
But if ; then a = 6, and b = 11, which gives the two roots both imaginary.
All equations whatever that have only two different powers of the unknown quantity, of which the index of the one is just double to that of the other, are resolved like Quadratics, by completing the square. Thus, the equation , by completing the square becomes ; whence, extracting the root on both sides, , where the root x has four values, because the given equation rises to the 4th power. See Equation.