Ex. 10. What are the values of x in the equation Ans. x=+3 or -7. (169.) The preceding method of solving a complete equation is applicable to all cases; nevertheless, it sometimes leads to inconvenient fractions. For, in order to reduce a given equation to the required form, X° +2px=9, we must divide by the coefficient of x®, which it is often impossible to do without a remainder. Let, then, the equation be represented by the form axʻ+bx=C. Multiply each member of the equation by 4a, and it becomes 4aRx* +4abx=4ac. Adding bi to both members, we have 4a’x? +4abx+69=4ac+b'. The first member of the equation is now a complete square, and its square root is 2ax+b. (170.) Hence, for completing the square, we may use the following RULE. Multiply the equation by four times the coefficient of x', and add to both sides the square of the coefficient of x. If the coefficient of xis unity, this rule becomes, Multiply the equation by four, and add to each member the square of the coefficient of x. , Either of these methods of completing the square QUEST.-What inconvenience sometimes results from the preceding method of solution? What method is free from this inconvenience ? may be practiced at pleasure; but the first niethod is to be preferred, except when its application would in. volve inconvenient fractions. Ex. 11. What are the values of x in the equation ——40=170. Transposing, we obtain x*— x=210. Multiplying by 4, 4x*— 4x=840. Adding 1 to each member of the equation, 4x*— 4x+1=841. Extracting the square root, 2x-1=+29. Whence, 2x=1=29=+30 or -28; . that is, . x=+15 or -14. Ex. 12 What are the values of x in the equation . 3x2+2x–9=76? Transweis.g, 3x*+2x=85 Multiplying each member wy 12, we have 360° +24x=1020. Adding the square of 2 to each member, we obtain 36x*+24x+4=1024. Extracting the square root, 6x+2=+32. Whence 6x=-2+32=+30 or -34; that is, x=+5 or -53.... Ex. 13. What are the values of x in the equation 22 X +20=42; ? Quest.-When is the first method to be preferred ? 32°—2x=133. Completing the square and extracting the root, 6x=2=40=42 or -38. Whence : x=7 or -6}. Ex. 14. What are the values of x in the equation XX - +78=8 ? Clearing of fractions, completing the square, and extracting the root, we have 24x=8=28=36 or -20. Whence x=1} or -5 Ex. 15. What are the values of x in the equation 4x*— 3x=85 ? Ex. 16. What are the values of x in the equation 35-33 u 6x+- =44 ? Ex. 17. What are the values of x in the equation 5.2 Ans. x=25 or 1. Ex. 18. What are the values of x in the equation 2x 1 7 Ex. 19. What are the values of x in the equation |