Question

Let x^2-mx+24 be a quadratic with roots x_1 and x_2. If x_1 and x_2 are integers, how many different values of m are possible?

Answer 1

An equation is a statement of equality „=‟ between two expression for particularvalues of the variable. For example5x + 6 = 2, x is the variable (unknown)The equations can be divided into the following two kinds:Conditional Equation:It is an equation in which two algebraic expressions are equal for particularvalue/s of the variable e.g.,a) 2x = 3 is true only for x = 3/2 b) x2 + x –  6 = 0 is true only for x = 2, -3 Note: for simplicity a conditional equation is called an equation.Identity:It is an equation which holds good for all value of the variable e.g;a) (a + b) xax + bx is an identity and its two sides are equal for all values of x. b) (x + 3) (x + 4) x2 + 7x + 12 is also an identity which is true for all values of x.For convenience, the symbol „=‟ shall be used both for equation and identity. 1.2 Degree of an Equation:The degree of an equation is the highest sum of powers of the variables in one of theterm of the equation. For example2x + 5 = 0 1st degree equation in single variable3x + 7y = 8 1st degree equation in two variables2x2  –  7x + 8 = 0 2nd degree equation in single variable2xy –  7x + 3y = 2 2nd degree equation in two variablesx3  –  2x2 + 7x + 4 = 0 3rd degree equation in single variablex2y + xy + x = 2 3rd degree equation in two variables1.3 Polynomial Equation of Degree n:An equation of the formanxn + an-1xn-1 + ---------------- + a3x3 + a2x2 + a1x + a0 = 0--------------(1)Where n is a non-negative integer and an, an-1, -------------, a3, a2, a1, a0 are realconstants, is called polynomial equation of degree n. Note that the degree of theequation in the single variable is the highest power of x which appear in the equation.Thus3x4 + 2x3 + 7 = 0x4 + x3 + x2 + x + 1 = 0 , x4 = 0are all fourth-degree polynomial equations.By the techniques of higher mathematics, it may be shown that nth degree equation ofthe form (1) has exactly n solutions (roots). These roots may be real, complex or amixture of both. Further it may be shown that if such an equation has complex roots,they occur in pairs of conjugates complex numbers. In other words it cannot have anodd number of complex roots.A number of the roots may be equal. Thus all four roots of x4 = 0are equal which are zero, and the four roots of x4  –  2x2 + 1 = 0Comprise two pairs of equal roots (1, 1, -1, -1)