Plotting the graph of x2 4x + 4 =y, Fig. II, we note that the curve touches the X axis at +2, but does not cross the axis. This indicates that the two roots of the equation are each equal to +2. 4x + 6 = 0. 3. Solve x2 Plotting the graph of x2 - 4x+6=y, Fig. III, we see that the graph approaches the X axis but does not cut it. This shows that the equation x2 + 4x + 6 O has no real roots, both roots being = imaginary, as may be shown by an algebraic solution. EX X Y FIG. IV. The graph of any equation of the form ax2 + bx + c = O as shown, is one of the curves produced by conic sections and is called a parabola. Note. When the roots of an equation are fractional, the graph will not intersect the X axis at a point of division and the values of the roots must be estimated approximately. In such cases, the graph should be drawn as accurately as possible. A modification of the foregoing method is sometimes used. 4. Solve x2 + x Let 6 Then y + x 6 = = 0. Using these two equations as a simultaneous system, the graph of equation (1) is a parabola, and that of equation (2) a straight line, Fig. IV. = The points of intersection of these two graphs, will locate the roots. The straight line cuts the parabola when x +2 and when x = -3. Hence these are the required roots. The advantage of this method lies in the fact that the same parabola x2 may be used for any number of equations. y = EQUATIONS INVOLVING TWO OR MORE RADICALS 141. In order to solve equations containing two or more radicals, practically the same methods are used as in the case of first degree equations involving radicals. The equation must be cleared of radicals, usually by raising the equation to the necessary power. If the equation contains two or more radicals, involution may be necessary two or more times to rationalize the equation. The student should note the importance of simplifying the equation before applying involution. 1. Solve √x+6 − √x + 1 = √x − 2. х Squaring, x+6 − 2√(x + 6) (x + 1) + x + 1 = x Add, to both sides, the square of half the original coefficient of x, Verify the solution, by substituting the values of x in the original As this equation does not reduce to an identity, - is not a root of this equation. In the solution of radical equations, involution may introduce factors into the equation that will produce values of x that do not satisfy the original equation. Hence the values found should always be substituted in the original equation to see whether they are roots. Values that do not satisfy the equation should be discarded. EQUATIONS IN THE QUADRATIC FORM 142. A complete quadratic equation contains the second and first power of the unknown number, that is, one exponent is twice the other. Hence, any equation that contains only two powers of the unknown number is said to be in the quadratic form, if one of the exponents is twice the other. Such equations may be solved by the methods used in solving quadratics. Completing the square by adding to each side the square of the quotient obtained by dividing the second term by twice the square root of the first term, -36 +25 = 18 4x4 25x2 + (25)2 == Extracting the square root, 2x2 Dividing by 2, Extracting the square root, x = +2 or + Hence this equation of the fourth degree has the four roots, +2, −2, + and 2. Solve 6x+6x1 = 13. Changing the sign of the exponent, 6 6x+ = 13 36x 78x + (13)2 -36 +169 - Squaring, = x = = x2-7x-14 + √x2 - 7x - 14 Substituting these values, 7x14, then y2 = x2 - 7x 9 or 4 墨 or養 20. = 6 14 or 3 or 2 |
