Quant02
Download as PPTX, PDF0 likes451 views
An equation is a mathematical statement of equality. There are different types of equations including simple equations with one unknown, simultaneous linear equations with two or more unknowns, and quadratic equations where the highest exponent is two. Simultaneous linear equations can be solved using elimination or cross-multiplication methods. The roots of a quadratic equation can be found using the quadratic formula and are impacted by the discriminant. Equations can also be applied to coordinate geometry, such as finding the distance between points and defining the equation of a straight line.
![Chapter 2. EquationsQuadratic EquationsAn 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 Equationx = [- b ± (b2 – 4ac)] / 2aSum of roots = - b / a = - (coefficient of x / coefficient of x2)Product of roots = c / a = (constant term / coefficient of x2)Construction of a Quadratic Equationx2 – (sum of roots) x + (product of roots) = 0Revision Notes – Quantitative Aptitudewww.cptsuccess.comPage 1 of 1](https://image.slidesharecdn.com/quant02-100413053319-phpapp02/85/Quant02-4-320.jpg)
![Chapter 2. EquationsApplication 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 constantThe Slope of the line is given by, m = (y2 – y1) / (x2 – x1)Revision Notes – Quantitative Aptitudewww.cptsuccess.comPage 1 of 1](https://image.slidesharecdn.com/quant02-100413053319-phpapp02/85/Quant02-6-320.jpg)
Quant02
- 1. Chapter 2. EquationsAn equation is defined to be a mathematical statement of equality. Simple equationA simple equation in one unknown x is in the formax + b = 0, where a, b are known as constants and a ≠ 0 A simple equation has only one root Simultaneous linear equations in two unknownThe 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 Aptitudewww.cptsuccess.comPage 1 of 1
- 2. Chapter 2. EquationsMethods to Solve Simultaneous linear equationsElimination 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 Aptitudewww.cptsuccess.comPage 1 of 1
- 3. Chapter 2. EquationsMethods to Solve Simultaneous linear equations: Cross-multiplication methodFor 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 methodsRevision Notes – Quantitative Aptitudewww.cptsuccess.comPage 1 of 1
- 4. Chapter 2. EquationsQuadratic EquationsAn 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 Equationx = [- b ± (b2 – 4ac)] / 2aSum of roots = - b / a = - (coefficient of x / coefficient of x2)Product of roots = c / a = (constant term / coefficient of x2)Construction of a Quadratic Equationx2 – (sum of roots) x + (product of roots) = 0Revision Notes – Quantitative Aptitudewww.cptsuccess.comPage 1 of 1
- 5. Chapter 2. EquationsRoots 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 equalIf b2 – 4ac > 0, the roots are real and distinct (unequal)If b2 – 4ac < 0, the roots are imaginaryIf b2 – 4ac is a perfect square the roots are real rational and distinctIf b2 – 4ac > 0 but not a perfect square the roots are real irrational and distinct Other propertiesIrrational roots occur in pairs. If p+ q is one root, then the other root p - qIf a = c then one root is reciprocal to the otherIf b = 0 the roots are equal but of opposite signsRevision Notes – Quantitative Aptitudewww.cptsuccess.comPage 1 of 1
- 6. Chapter 2. EquationsApplication 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 constantThe Slope of the line is given by, m = (y2 – y1) / (x2 – x1)Revision Notes – Quantitative Aptitudewww.cptsuccess.comPage 1 of 1