QUADRATICS AND BEYOND CHAPTER VIII QUADRATIC EQUATIONS 88. Definition. An equation that contains the second but no higher power of the variable is called a quadratic equation. The most general form of the quadratic equation in one variable is where we shall always assume a, b, and c to represent rational numbers, and where a 0. Every quadratic equation in x can be brought to this form by transposing and simplifying. 89. Solution of quadratic equations. The solution of a quadratic equation consists in finding its roots, that is, the numbers (or expressions involving the coefficients in case the coefficients are literal) which satisfy the equation. The common method of solving a quadratic equation consists in bringing the member of the equation that involves the variable into the form of a perfect square, i.e. into the form If now we add 1 to both sides of the equation, the left-hand member will be a perfect square, x2 + 2 x + 1 = 9. Express as a square, (x + 1)2 = 9. x=- 4 or 2. Extract the square root, x+1=±3. Transpose, Both 4 and 2 satisfy the equation, as we see on substituting them for x. Thus to both members to make the left-hand member a Let us solve Transpose c, Divide by a, 2 That the equation can have no other roots appears from § 96. may be used as a formula for the solution of a quadratic equation. we may substitute in the formula a=2, b=-3, c= 6, and obtain 1 X 2 = One should verify the fact that both − b −√b2 — 4 ac 2 a -- b + √b2 - 4 ac and 2 a satisfy (1) and are consequently roots of the equation. They are, in general, distinct from each other. For particular values of the coefficients to be noted later (§ 98) the roots may be equal or complex (i.e. of form a +BV-1, where a and B are ordinary rational or irrational numbers). We may sum up the process of solving a quadratic equation in the following RULE. Write the expression in the form ax2 + bx + c = 0. Transpose the term not involving x to the right-hand side of the equation. Divide both sides of the equation by the coefficient of x2. Add to both members the square of one half of the coefficient of x, thus making the left-hand member a perfect square. Rewrite the equation, expressing the left-hand member as the square of a binomial and the right-hand member in its simplest form. Extract the square root of both members of the equation, not omitting the sign in the right-hand member. Transpose the constant term, leaving x alone on the left-hand side of the equation. The two values obtained on the right-hand side by taking the + and signs separately are the roots sought. Check by substituting the solutions in the original equation, which should then reduce to an identity. 90. Pure quadratics. A quadratic equation in which the coefficient of the term in x is zero is often called a pure quadratic. Its solution is found precisely as in the general case, excepting that we do not need to complete the square. Thus let us solve |