Here 7 cannot be found exactly; but we can find an approximate value of it to any assigned degree of accuracy, and thus obtain the value of x to any assigned degree of accuracy. 321. In the examples hitherto considered we have found two different roots of a quadratic equation; in some cases however we shall find really only one root. Take for example the equation x2 - 12x+36= 0; by extracting the square root we have x 6 = 0, and therefore x = 6. It is however convenient in this case to say that the quadratic equation has two equal roots. 322. If the quadratic equation be represented by ax2 + bx + c = 0, we know from Art. 318 that the two roots are respectively Now these will be different unless b2-4ac = 0, and then each of them is b 2a This relation b2 - 4ac = 0 is then the condition that must hold in order that the two roots of the quadratic equation may be equal. 323. Consider next the example x2 - 10x + 32 = 0. By transposition, x10x=-32; by addition, x2 - 10x + 25 = 25 – 32 = -7. If we proceed to extract the square root we have But the negative quantity -7 has no square root either exact or approximate (Art. 232); thus no real value of x can be found to satisfy the proposed equation. In such a case the quadratic equation has no real roots; this is sometimes expressed by saying that the roots are imaginary or impossible. We shall return to this point in a subsequent chapter. See Chapter XXV. 324. If the quadratic equation be represented by ax2 + bx + c = 0, we see from Art. 318 that the roots are real if b2 - 4ac is positive, that is, if b2 is algebraically greater than 4ac, and that the roots are impossible if b2-4ac is negative, that is, if b3 is algebraically less than 4ac. 20. (x-1)(x-2)+(x-2) (x-4)=6 (2x-5). 22. (5x-3)-7=44x+5. |