Form 2 Mathematics – QUADRATIC EQUATION msomimaktaba, November 13, 2018August 17, 2024 QUADRATIC EQUATION QUADRATIC EQUATIONIs any equation which can be written in the form of ax2 + bx + c=0 where a ≠ 0 and a, b and c are real numbers.SOLVING QUADRATIC EQUATIONi) BY FACTORIZATIONExample 1solve x2 + 3x – 10 = 0Solution:x2 + 3x – 10 = 0(x2 – 2x) + 5 (x – 2) = 0x (x – 2) + 5 (x – 2) = 0(x + 5) (x – 2) = 0Now x + 5 = 0 or x – 2 = 0x = -5 or x = 2x = -5 or 2 Example 2Solve for xi) 2x2 + 9x + 10 = 0 Solution:Sum = 9Product = 2 x 10 = 2020 = 1 x 20= 2 x 10= 4 x 5(2x2 +4x) + (5x + 10) = 02x(x + 2) + 5(x + 2) = 0(2x + 5) (x + 2) = 0Now,2x + 5 = or x + 2 = 0x= -2.5 or -2 ii) 2x2 – 12x = 0Solution:2x(x – 6) = 02x = 0 or x – 6 = 0X = 0, or x = 6X = 0 or 6 iii) x2 – 16 = 0Solution:x2 – 16 = 0(x2) – (4)2 = 0(x + 4) (x – 4) = 0Now, x + 4 = 0 or x – 4 = 0x = -4 or x = 4 EXERCISE1. Solve for x fromX2 – 7x + 12=0Solution:x2 – 3x – 4x + 12 = 0(x2 – 3x) – (4x – 12) = 0x(x – 3) – 4(x – 3) = 0(x – 4) (x – 3) = 0Now, x – 4 = 0 or x – 3 = 0x= 4 or x = 3 ii) 4x2 – 20x + 25 = 0 Solution:4x2 – 10x – 10x – 25 = 0(4x2 – 10x) – (10x – 25) = 02x(2x – 5) – 5(2x – 5) = 0(2x – 5) (2x – 5) = 0Now, 2x – 5 = 0 or 2x – 5 = 0x = iii) 4x2 – 1 = 0Solution:4x2 – 1 = 022x2 – 1 = 0(2x)2 – (1)2 = 0(2x + 1) (2x – 1) = 0Now, 2x + 1 = 0, or 2x – 1 = 0X = or x =iv) (x – 1)2 – 81 = 0Solution:(x – 1)2 – 92 = 0(x – 1 – 9)(x – 1 + 9) = 0Now, x – 1 – 9 = 0, or x – 1 + 9x – 10 = 0, x + 8 = 0x = 10 or x = – 8v) 2x2 = 10xSolution:2x2 – 10x = 02x(x – 5) = 02x = 0 or x – 5 = 0x = 0, or x = 5 SOLVING BY COMPLETING THE SQUAREExample 1Solve i) 2x2 + 8x – 24 = 0Solution:x2 + 4x – 12 = 0x2 + 4x = 12x2 + 2x + 2x + 4 = 12 + 4(x2 + 2x) + (2x + 4) = 16x(x+ 2) + 2(x +2) = 16(x +2) (x +2) = 16(x +2)2 = 16= x + 2 = 4X = 4 2X = 2 or x =6X = 2 or 6 ii) x2 + 5x – 14 = 0solution:x2 + 5x = 14(x2 + ) + (+ ) = 14 + x(x + ) +(x + ) = (x + )(x + ) = = x + = x= or x = x = 2 or 7iii) 3x2 – 7x– 6 = 0Solution:x2 – – 2 = 0x2 – = 2x2 – – + = 2 + (x2 – ) –( – )= x(x – ) – (x– )= (x– )(x– )= = x – = Now,x – = , x – = x = 3 or x = iv) x2 – 5x + 2 = 0x2 – 5x = -2x2 – – + = -2 + x(x – –) – (x – –) = (x – )2 = = x – = ±x = ± x = or GENERAL FORMULA 1. Solve ax2 + bx + c = 0Solution:x2 + + = 0x2 + =x2 + + + = + (x2 + ) +( + ) = x(x + ) + ( x + ) = (x + )2 = = x + = x = Generally, Example 1.Solve for x by using generally formulai) 6x2 + 11x + 3 = 0Solution: a = 6, b = 11, c = 3From the general equation, and and ii) 5x2 – 6x – 1 = 0Solution:a= 5, b = -6, c =1From the general equation, and and iii) 0 = 400 + 20t – t2solution:t2 20t 400 = 0a = 1, b = -20, c = -400From the general equation GRAPHICAL SOLUTION OF QUADRATIC EQUATION– The general quadratic equation ax2 + bx + c =0 can be solved graphically– First draw the graph by setting ax2 + bx + c = y and thenDrawing graphsExample 1Draw the graph of the following equationi) y = x2 – 3ii) y = 2 – x2iii) y = x2 + x – 1Solution:i) y = x2 – 3TABLE VALUE x-3-2-10123y61-2-3-216 ii) y = 2 – x2 iii) y = x2 + x – 1Solution: APPLICATION OF GRAPHS IN SOLVING QUADRATIC EQUATIONa) Solve graphically the equation x2 – x – 6 = 0b) Use the graph in a to solve the equationx2 – x – 2 = 0Solution: x2 – x – 6 = 0Let y=x2 – x – 6…………………….(i) Then y=0………………….(ii) (b)From x2 – x – 6 = 0 Then x2 – x – 2 = 0 can be written as x2 – x –2-4 = 0-4 x2 – x – 6 = -4 But y=x2 – x – 6 ∴y=-4∴x=-1 or x=2More examples1. A man is 4 times as old as his son. In 4 years the product of their ages will be 520.Find the sons present ageNow(x + 4) (4x + 4) = 5204x2 + 4x + 16x + 16 = 520+ – = x2 + 5x – 126 = 0a=1, b = 5, c = -126From the general equation, and and x = 9 or -14.The present age of the son is 92.Find the consecutive numbers such that the sum of their squares is equal to 145Solution:Let x be the first number and x + 1 be the second numberSum of x2 + (x + 1)2 = 145 their squaresNow, x2 + (x + 1)2 = 145x2 + x2 + 2x + 1 = 1452 x2 + 2x – 144 = 0Divide by 2 both sides , then x2 + x – 72 a =1, b =1, c = -72From the general equation, and and x = 8 and 9or x = -9 and -8The two consecutive numbers are 8 and 9 or -9 and -8. ALL NOTES FOR ALL SUBJECTS QUICK LINKS:AGRICULTURE O LEVEL PURE MATHEMATICS A LEVELBAM NOTES A LEVELBASIC MATH O LEVELBIOLOGY O/A LEVELBOOK KEEPING O LEVELCHEMISTRY O/A LEVELCIVICS O LEVELCOMPUTER(ICT) O/A LEVELECONOMICS A LEVELENGLISH O/A LEVELCOMMERCE O/A LEVELACCOUNTING A LEVELGENERAL STUDIES NOTESGEOGRAPGY O/A LEVELHISTORY O/A LEVELKISWAHILI O/A LEVELPHYSICS O/A LEVELMOCK EXAMINATION PAPERSNECTA PAST PAPERS Basic Mathematics Study Notes Form 2 Basic Mathematics Study Notes Msomi Maktaba All Notes FORM 2MATHEMATICSPost navigationPrevious postNext postRelated Posts Form 2 Kiswahili Study Notes Form 2 Kiswahili – UFAHAMU November 13, 2018February 13, 2019Please Search Your Topic From The Document Below ALL NOTES FOR ALL SUBJECTS QUICK LINKS: AGRICULTURE O LEVEL PURE MATHEMATICS A LEVEL BAM NOTES A LEVEL BASIC MATH O LEVEL BIOLOGY O/A LEVEL BOOK KEEPING O LEVEL CHEMISTRY O/A LEVEL CIVICS O LEVEL COMPUTER(ICT) O/A LEVEL ECONOMICS A LEVEL ENGLISH… Read More English Language Study Notes LANGUAGE ONE FORM 5 – INTRODUCTION TO LANGUAGE November 11, 2018May 1, 2020ALL NOTES FOR ALL SUBJECTS QUICK LINKS: AGRICULTURE O LEVEL PURE MATHEMATICS A LEVEL BAM NOTES A LEVEL BASIC MATH O LEVEL BIOLOGY O/A LEVEL BOOK KEEPING O LEVEL CHEMISTRY O/A LEVEL CIVICS O LEVEL COMPUTER(ICT) O/A LEVEL ECONOMICS A LEVEL ENGLISH O/A LEVEL COMMERCE O/A LEVEL ACCOUNTING A LEVEL… Read More Form 3 Physics Notes PHYSICS FORM THREE TOPIC 7: MEASUREMENT OF THERMAL ENERGY November 6, 2018February 13, 2019 Heat Capacity Heat capacity is the amount of heat required to raise the temperature of an object or substance by one degree. 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