Completing the square, 16 (2x2 - x)2 — 12 (2x2-x)+ =33+ Extracting the square root of each side, (2x2-x) This Equation will give 14 ± 53. Find the value of x in the Equation x38x2+19x=12. 2x2-x-6; whence x= = 2, or -14. Transposing the 12, and multiplying by x, we have x48x3+19x2-12x=0. 23 4 We have now a Biquadratic Equation, which may be reduced to the form. of a Quadratic by the method just exemplified. x1 — 8x3+19x2 — 12x (x2 —4x We find the first two terms of the square root of the first member, by the common rule. Completing the square, (x2 —4x)2+3(x2 — 4x)=0. 1 16 3x2-12x. This remainder may be resolved into 3(x2—4x), in which the binomial factor is the same as the part of the square root above found. Then we shall have the Equation 9 4 = x2-4x=-3, or 0. 3 ૭! 529 16 The Equation 22-4x=-3 will give x=3, or 1. 54. Find the value of x in the Equation x42x3+x=5112. The first two terms of the square root of the first member are x2-x; and the remainder is x2+x; which may be put under the form -(x2-x). Hence, we shall have the Equation (x2-x)2-(x2-x)=5112.. -- 143 2 x+2x3-7x2-8x=-12. The first two terms of the square root of the first member are x2+x; Completing the square, we find and the remainder is 8x2-8x; which may be expressed by -8(x2+x). We shall then have the Equation (x2+x)2-8(x2+x)=-12. 20449 4 (x2+x)2-8(x2+x)+16=4. Extracting the square root of each side (x2+x)-4=2; hence x2+x=6, or 2. == The Equation x2+x=6 will give x=2, or -3. - 56. Find the value of x in the Equation x10x3+35x2—50x+24=0. Transposing, x4-10x3+35x2-50x——24. The first two terms of the square root of the first side of the Equa tion are x–5; and the remainder is 10x2-50x; which may be expressed by 10(2−5x). Hence, we have the Equation (2_5)+10(2−5)=−24. Completing the square, (–5)+10(–5)+-25=1. Extracting the square root, –52+5=±1. From the Equation x2-5x+5=+1, x will be found =4, or 1. From the Equation x2-5x+5=-1, x will be found 3, or 2. 57. Find the value of x in the Equation x4-12x3+44x2-48x=9009. The first two terms of the square root of the first number are x2-6x; and the remainder is 8x2-48x; which may be put under the form 8(x2-6x). Hence, we shall have the Equation (x2-6x)2+8(x2-6x)=9009. Completing the square, we have (x2-6x)2+8(x2-6x)+16=9025. Extracting the square root of each side 58. Find the value of x in the Equation (x2-6x)+495. From the Equation x2-6x+4=+95, we shall find x=13, or —7. x4-8ax3+8a2x2+32a3x=s. The first two terms of the square root of the first member of the Equation are x2-4ax; and the remainder is 8a2x2+32a3x; which may be expressed by -8a2(x2-4ax). 6 Hence, we shall have (x2-4ax)2-8a2(x2 —4ax)=s. Completing the square, we find (x2-4ax)2 — 8a2(x2 — 4ax)+16a2=s+16a1. Extracting the square root of both sides, (x2-4ax)-4a2±√s+16a. Transposing -4a2, we have x2-4ax=4a2±√/s+16a2. Completing the square, x2-4ax +4a2=8a2±√/3+16a1. Extracting the square root, x-2a= 8a2±√/s+16a1. Transposing-2a, we find x=2a±√/8a2±√/s+16a2. PROBLEMS In Pure Equations and Affected Quadratics Containing but One Unknown Quantity. X x2 1. Find two numbers such that their product shall be 750, and the quotient of the greater divided by the less, 31. Let x represent the greater number; then 750 represents the less; and represents the quotient of the greater divided by the less. 750 We shall then have the Equation x2 =3}, 750 from which we shall find x=50, the greater; ther =15, is the less. 750 50 2. Find a number such that if and of it be multiplied together, and the product divided by 3, the quotient will be 298. Let a represent the number; then x2 56 x2 represents the product of and of the number; and represents this product divided by 3. 168 We shall then have the Equation x2 168 from which we shall find x=224. 9 4x2 =2983, 3. A mercer bought a piece of silk for £16 4s.; and the number of shillings that he paid per vard, was to the number of yards, as 4 to 9. How many yards did he buy? and what was the price per yard? Let a represent the number of yards he bought; then 4x represents the number of shillings per yard; and represents the number of shillings the whole cost; also 9 £16 4s. is equal to 324 shillings. Hence, we shall have the Equation 4x2 9 12 shillings was the price per yard. = =324; whence x=27 yards. Then of 27 4. Find two numbers which shall be to each other as 2 to 3, and the sum of whose squares shall be 208. Let x represent the greater number; then 2x 3 represents the less number; and the Equation will be 4x2 9 x2+ =208; which will give x=12, the greater number; hence of 12=8, is the less number. 5. A person bought a quantity of cloth for $120; and if he had bought 6 yards more for the same sum, the price per yard would have been $1 less. What was the number of yards? and the price per yard? |