1. Chapter 2. Equations An equation is defined to be a mathematical statement of equality. Simple equation A simple equation in one unknown x is in the form ax + b = 0, where a, b are known as constants and a ≠ 0 A simple equation has only one root Simultaneous linear equations in two unknown The general form of a linear equation in two unknowns x and y is ax + by + c = 0 where a and b are non-zero coefficients. Two equations a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 form a pair of simultaneous equations in x and y. A value for each unknown which satisfies both equations at the same time gives the roots / solution of the equation. Revision Notes – Quantitative Aptitude www.cptsuccess.com Page 1 of 1

2. Chapter 2. Equations Methods to Solve Simultaneous linear equations Elimination method: In this method one unknown is eliminated, thus reducing two linear equations to a linear equation in one unknown. This unknown is solved and its value substituted in the equation to find the other unknown. Revision Notes – Quantitative Aptitude www.cptsuccess.com Page 1 of 1

3. Chapter 2. Equations Methods to Solve Simultaneous linear equations: Cross-multiplication method For two equation, a1x + b1y + c1 = 0, and a2x + b2y + c2 = 0 Coefficients of x and y and constant term are arranged as: which gives: x / (b1 c2 – b2 c1) = y / (c1 a2 – c2 a1) = 1 / (a1 b2 – a2 b1) Hence, x = (b1 c2 – b2 c1) / (a1 b2 – a2 b1) y = (c1 a2 – c2 a1) / (a1 b2 – a2 b1) Equations in three variables can also be solved using the above two methods Revision Notes – Quantitative Aptitude www.cptsuccess.com Page 1 of 1

4. Chapter 2. Equations Quadratic Equations An equation in the form ax2 + bx + c = 0, where x is a variable and a, b and c are constants with a ≠ 0 is called a quadratic equation. When b = 0 the equation is called a pure quadratic equation and when b ≠ 0 the equation is called an affected quadratic. Roots of a Quadratic Equation x = [- b ±  (b2 – 4ac)] / 2a Sum of roots = - b / a = - (coefficient of x / coefficient of x2) Product of roots = c / a = (constant term / coefficient of x2) Construction of a Quadratic Equation x2 – (sum of roots) x + (product of roots) = 0 Revision Notes – Quantitative Aptitude www.cptsuccess.com Page 1 of 1

5. Chapter 2. Equations Roots of a Quadratic Equation b2 – 4ac is known as the discriminant in the equation as it discriminates the nature of roots of the equation If b2 – 4ac = 0, the roots are real and equal If b2 – 4ac > 0, the roots are real and distinct (unequal) If b2 – 4ac < 0, the roots are imaginary If b2 – 4ac is a perfect square the roots are real rational and distinct If b2 – 4ac > 0 but not a perfect square the roots are real irrational and distinct Other properties Irrational roots occur in pairs. If p+ q is one root, then the other root p - q If a = c then one root is reciprocal to the other If b = 0 the roots are equal but of opposite signs Revision Notes – Quantitative Aptitude www.cptsuccess.com Page 1 of 1

6. Chapter 2. Equations Application of Equations to Coordinate Geometry Distance of a point P (x, y) from Origin (0, 0) is  (x2 + y2) Distance between two points P (x1, y1) and Q (x2, y2) is  [(x1 – x2)2 + (y1 – y2)2] Equation of a straight line is written as y = mx + c, where m is the slope and c is the constant The Slope of the line is given by, m = (y2 – y1) / (x2 – x1) Revision Notes – Quantitative Aptitude www.cptsuccess.com Page 1 of 1