...becomes equal to p2. Hence, 3d. The greatest possible value of the product of the two roots, is equal to the square of half the co-efficient of the first power of the unknown quantity. • Of the Four Forms. 117i Thus far, we have regarded p and q as algebraic quantities, without considering...

...first power of the unknown quantity, add and subtract the square root of the known term increased by the square of half the coefficient of the first power of the unknown quantity, and the results will be the two values of x. In many of the following solutions the processes of completing...

...taken with a contrary sign, plus or minus the square root of the known term after it has been added to the square of half the coefficient of the first power of the unknown quantity. REMARK. — The student should apply this rule in the solution of the following examples after they...

...becomes equal to p*. Hence, 3d. The greatest possible value of the product of the two roots, is equal to the square of half the co-efficient of the first power of the unknown quantity. Of the Four Forms. 117. Thus far, we have regarded p and q as algebraic quantities, without considering...

...But a, whose square, (a2), we have added, is half the coefficient of x in the second term. Hence, If the square of half the coefficient of the first power of the unknown quantity be added to both members of a quadratic equation in the form of x2 + 2аж = b, the first member will...

...impossible (Art. 193). Hence, when q is greater than p1, that is, when the known term is negative, and greater than the square of half the coefficient of the first power of x, the roots art imaginary. To show why the roots are imaginary, we must inquire, into what two parts...

...In the third and fourth forms, when q is greater thanp2, that is, when the known term is negative, and greater than the square of half the coefficient of the first, power of X, both values of the unknown quantity are impossible. To explain the cause of this impossibility,...

...negative, and the extraction of the root is impossible, (Art. 193.) Hence, When the known term is negative, and greater than the square of half the coefficient of the first power of x, the roots are imaginary. To show why the roots are imaginary, we must prove that When a number is...

...unknown quantity, its value is found by transposing the second term. SCH. 1. — This process of adding the square of half the coefficient of the first power of the unknown quantity to the first member, in order to make it a perfect square, is called COMPLETING THE SQUARE. There are...

...But a, whose square (a8) we have added, is half the coefficient of x in the second term. Hence, V If the square of half the coefficient of the first power of the unknown quantity be added to both members of a quadratic equation in the form of a? + Zaz = b, the first member will...