7. What are the values of x in the equation 8. What are the values of x in the equation 10. What are the values of x in the equation 11. What are the values of x in the equation 1. What are the values of x in the equation x2- 8 x = — 7? Completing the square, we have that is, in this form both the roots are positive. VERIFICATION. If we take the greater root, the equation Hence both of the roots will satisfy the equation. 2. What are the values of x in the equation — 1 x2 + 3 x − 10 = 14x2-18x+40? Clearing of fractions, we have 2 Hence -6, we have x=3.5+1.5 = 5, and x = 3.5 — 1.5 = 2. VERIFICATION. If we take the greater root, the equation 3. What are the values of x in the equation 3x+2x2+1= 17 x 2x2 - 3? Hence x′ = 24 + 12 −5, and x'= 23-12-1 5 = = 5 5 VERIFICATION. If we take the greater root, the equation 4. What are the values of x in the equation 5. What are the values of x in the equation 6. What are the values of x in the equation x' = 3. 1 22 7. What are the values of x in the equation 8. What are the values of x in the equation +100=2x2+12x-26? Ans. {x" = 6. x' = 7. 5 5 = 9. What are the values of x in the equation 10. What are the values of x in the equation 161. We have seen (§ 153) that every complete equation of the second degree may be reduced to the form Now, since +2px+p2 is the square of x+p, and q+p2 the square of Va+p, we may regard the first member as the difference between two squares. Factoring (§ 56), we have (x+p+√q+p3)(x+p−√q+p2) = 0 (3) This equation can be satisfied only in two ways, first, by attributing such a value to x as shall render the first factor equal to 0; or, second, by attributing such a value to x as shall render the second factor equal to 0. Placing the second factor equal to 0, we have x + p −√q+p2 = 0; and x'=-p+√q+p* (4) Placing the first factor equal to 0, we have x+p+√q+p2 = 0; and x"=-p-√q+p2 (5) Since every supposition that will satisfy Equation (3) will also satisfy Equation (1), from which it was derived, it follows that x' and x" are roots of Equation (1); also that Every equation of the second degree has two roots, and only two. NOTE.. The two roots denoted by x' and x'' are the same as found in 2158. Second Property. 162. We have seen (§ 161) that every equation of the second degree may be placed under the form (x+p+ √q+p2)(x+p−√q+p3) = 0. By examining this equation, we see that the first factor may be obtained by subtracting the second root from the unknown quantity x; and the second factor, by subtracting the first root from the unknown quantity x. Hence Every equation of the second degree may be resolved into. two binomial factors of the first degree; the first terms in both factors being the unknown quantity, and the second terms, the roots of the equation taken with contrary signs. Third Property. 163. If we add Equations (4) and (5), § 161, we have |