Hence, when bz 4ac equals zero, the roots are equal. The expression b2 - 4ac is called the discriminant of the quadratic equation ax2 + bx + c = 0. 4. To determine the signs of the roots. When a is positive, b and c either positive or negative, the signs of the roots rı and r2 may be determined from the signs of b and c. If c is positive, Vb2 - 4ac < b, hence both roots have the sign of -b. If C is negative, V62 – 4ac > b, and the roots will have opposite signs, since rı will be positive and r2 negative. If b is negative, - b is positive and therefore rı will be numerically greater than r2. If b is positive, - b is negative and therefore rz will be numerically greater than ri. In general, if c is negative, the roots have opposite signs and the numerically greater root has the sign opposite to that of b. Determine the character of the roots of 2x2 + 3x 5. Reduce the equation to the general form, 2x2 + 3x – 5 = 0 hence a = 2 b = 3 62 4ac 9. + 40 = 49 5 As 62 4ac is a perfect square, the roots are real and rational. As 62 4ac not equal to zero, the roots are unequal. As c is negative, the roots have opposite signs, and as b is positive, the negative root is the numerically greater. с EXERCISE 135 Without solving, determine the character of the roots of: 1. 3x2 2x 8. 7. 7x2 + 5x 150. 2. 5x2 6x 27. 8. 6x2 + 5x 14. 3. 2x2 5x 7. 9. 8x2 + 7x 51. 4. 3x2 + 7x 6. 10. 7.x2 20x 5. 5x2 24. 11. 11x2 10x 24. 6. 8x2 + 3x 26. 12. 4x2 23x + 30 = 0. 75. 7x -4 have 13. For what values of m wil x2 + 2mx equal roots ? In the equation, x2 + 2mx + 4 = 0, When 62 4ac 0, the roots are equal. a = 1 b 2m Therefore 4m2 16 0 4m2 + 16 14. For what values of a will the equation 2x2 + 3ax + 2 = 0, have equal roots? 15. For what values of m will the equation 4x2 + (6 – 2ax = -1 have equal roots? 16. For what values of m will the equation (m + 1)x2 + (m – 1x + m + 1 = 0 have imaginary roots? RELATION OF ROOTS AND COEFFICIENTS 185. Any quadratic equation in the form ax2 + bx + c = 0, may be reduced to the form, x2 + px + y = 0, by dividing by the coefficient of x?. By solving x2 + px + q = 0, we find -p+ Vp 49 -p - Vp? – 49 2 2 2p by addition, ri + ra = - -P 2 p? – (p? – 49) by multiplication, rir2 = 9. 4 2 and 12 Hence, the sum of the roots of a quadratic equation in the form x2 + px + q = 0, is equal to the coefficient of x with its sign changed, and the product of the roots is equal to the absolute term. As the above principle corresponds exactly to the rule for writing the product of two binomials, the equation x2 + px + y = 0, may be written (x – ri) (x – r2) = 0. Therefore, when the roots are known, the equation may be formed by subtracting each root from x and placing the product of the remainders equal to zero, EXERCISE 136 1. Form an equation whose roots are (-1 + 1–2). Subtract each root from x, and write the product equal to zero. (x + 1 V-2)(x +1 + V-2) = 0 (x + 1)2 – (V-2)? 0 x2 + 2x + 3 = 0 Form the equation whose roots are: 2. 3,5. 9. a, b. 3.1, . 10. Va+b, Va – b. 4. 6, -8. 11. 1(2 + V3). 5. -2, -3. 12. 2 + V-3. 2 - V-3. 6. -3, -4. 13. a(1 + V3). a, - 3a. 14. (m + n), (m – n). 8. 2 – a, 2 + a. 15. }(3 + 5V2). GRAPHS OF HIGHER EQUATIONS 186. The graphs of equations or functions of a higher degree than the second, may be plotted by the methods already shown. This graphical method furnishes a convenient method of locating the real roots of higher equations. In general, the number of real roots of an equation in x will be equal to the number of times the graph cuts the X axis. When the graph touches the X axis without crossing it, that is, when the graph is tangent to the X axis, equal roots are indicated. If the graph approaches the X axis and then recedes without touching the axis, the roots are imaginary. FRANCISCUS VIETA His In Artem Analyticam Isagoge, 1591, is the first work with a symbolic treatment of algebra. Vieta arrived at an incomplete understanding of the relations between the coefficients and roots of an equation. He also wrote on geometry and trigonometry. EXERCISE 137 1. Locate the roots of x3 3x2 + 4x 2 = 0. Let y = x3 – 3x2 + 4x – 2, then any value of x that makes y 0 will be a root of the equation; hence any point of the graph that lies on the X axis indicates a root of the equation. For any negative value of x, y is negative, hence the curve cannot touch the X axis on the left of the Y axis, and as for positive values of x greater than 2, the value of y becomes positively greater, the graph goes away from the X axis. Hence, there is only one real root, : 1, the other two roots of this cubic equation are imaginary. (Fig. I.) |