Thus, a+x=√(a2+xy/c2+x2), to find x. 3d. When there are two radical expressions, it is generally better to make one of them stand alone on one side, before squaring. Thus, √(x-5)—3—4—√(x—12), to find x. Transposing, (x-5)=7-√(x-12). Squaring, x-5-49-14√(x-12)+x-12. Squaring, x-12=9, from which x=21. EQUATIONS OF THE SECOND DEGREE. ART. 206.—An Equation of the Second Degree (See Art. 148), is one in which the greatest exponent of the unknown quantity is 2. Thus, x2-9, and 5x2+3x=26, are equations of the second degree. An equation containing two or more unknown quantities, in which the greatest exponent, or the greatest sum of the exponents of the unknown quantities, is 2, is also an equation of the second degree. Thus, xy=6, x2+xy=8, xy+x+y=11, are equations of the second degree. Equations of the Second Degree, are frequently denominated Quadratic Equations. ART. 207.-Equations of the second degree are of two kindsincomplete and complete. An incomplete equation of the second degree, is of the form ax2-b, and contains only the second power of the unknown quantity, and known terms. Thus, x2=9, and 8x2-5x2-12, are incomplete equations of the second degree. An incomplete equation of the second degree, is frequently denominated a pure quadratic equation. A complete equation of the second degree, is of the form ax2+bx=c, and contains both the first and second powers of the unknown quantity, and known terms. Thus, 3x2+4x=20, and ax2—bx2+ dx—ex=f-g, are complete equations of the second degree. A complete equation of the second degree, is frequently denominated an affected quadratic equation. REVIEW.-206. What is an equation of the second degree? Give examples. If an equation contains two unknown quantities, when is it of the second degree? Give examples. 207. How many kinds of equations of the second degree are there? What are they? What is the form of an incomplete equation of the second degree? What does it contain? Give an example. What is the form of a complete equation of the second degree? What does it contain? Give an example. What is a pure quadratic equation? What is an affected quadratic equation? ART. 208.-Every equation of the second degree, may be reduced to one of the forms ax2=b, or ax2+bx=c. For, in an incomplete equation, all the terms containing a2 may be collected together, and then, if the coëfficient of x2 contains more than one term, it may be assumed equal to a single quantity, as a, and the sum of the known quantities, to another quantity, b, and then the equation becomes ax2-b, or ax2-b—0. So a complete equation may be similarly reduced; for all the terms containing x2, may be reduced to one term, as ax2; and those containing x, to one, as bx; and the known terms to one, as c; then the equation is ax2+bx=c, or ax2+bx-c=0. Hence, we infer: That every equation of the second degree, may be reduced to an incomplete equation involving two terms, or to a complete equation involving three terms. Frequent illustrations of these principles will occur hereafter. INCOMPLETE EQUATIONS OF THE SECOND DEGREE. ART. 209.-1. Let it be required to find the value of x in the equation 22-16-0. Transposing, x2-16 Extracting the square root of both members, Verification. x=4, that is, x=+4, or 4. (+4)2—16=16-16-0. or, (-4)2-16=16-16=0. 2. Find the value of x in the equation 5x2+4=49. 4. Given ax2+b=cx2+d, to find the value of x. From the preceding examples, we derive the FOR THE SOLUTION OF AN RULE, INCOMPLETE EQUATION OF THE SECOND Reduce the equation to the form ax2=b. Divide both sides by the coefficient of x2, and then extract the square root of both members. ART. 210.-If we take the equation ax2=b By separating into factors, (x+m)(x—m)=0. Now, this equation can be satisfied in two ways, and in two only; that is, by making either of the factors equal to 0. By making the second factor equal to 0, we have x-m=0, or x=- =+m. By making the first factor equal to 0, we have x+m=0, or x=-m. Since the equation (x+m)(x—m)=0, can be satisfied only in these two ways, it follows, that the values of x obtained from these conditions, are the only values of the unknown quantity. Hence we conclude 1st. That every incomplete equation of the second degree, has two roots, and only two. 2d. That these roots are equal, but have contrary signs. Find the roots of the equation, or the values of x, in each of the following examples. REVIEW.-208. To what two forms may every equation of the second degree be reduced? Why? 209. What is the rule for the solution of an incomplete equation of the second degree? 210. Show that every incomplete equation of the second degree, has two roots, and only two; and that those roots are equal, but have contrary signs. QUESTIONS PRODUCING INCOMPLETE EQUATIONS OF THE SECOND DEGREE. ART. 211.-In the solution of a problem producing an equation containing the second power of the unknown quantity, the equation is found on the same principle, as in questions producing equations of the first degree. See Art. 156. 1. Find a number, whose to 60. Let x the number; then multiplied by its, will be equal 2. What number is that, of which the product of its third and fourth parts is equal to 108? Ans. 36. 3. What number is that, whose square diminished by 16, is equal to half its square increased by 16? 4. What number is that, whose square diminished equal to the square of its half, increased by 54? Ans. 8. by 54, is Ans. 12. 5. What number is that, which being divided by 9, gives the same quotient, as 16 divided by the number? Ans. 12. 6. What two numbers are to each other as 3 to 5, and the dif ference of whose squares is 64 ? Let 3x the less number; then 5x the greater. And Or From which (5x)-(3x)2-64 25x2-9x2-16x2-64. x=2; hence, 3x=6 and 5x=10, are the numbers. See general directions, page 127. REVIEW.-211. In the solution of a problem producing an equation containing the second power of the unknown quantity, upon what principle is the equation found? |