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Additional mathematics/ pure mathematics ✓ nature of roots of a quadratic equation consider the quadratic equation ax² + bx + c = 0 Generally a quadratic equation is given by x² - (sum of roots)x + (product of roots) = 0 Letting the coefficient of x² inthe quadratic equation ax² + bx + c=0 to be 1 Our new equation will be x² +(b/a)x + c/a = 0 Equating to the general equation above you see that (sum of roots) = –b/a (product of roots) = c/a given that the roots of a quadratic equation are ohm an mhü respectively Ω an μ hence Ω + μ = -b/a Ω × μ = c/a *Some simple identities* Ω² + μ² = (Ω+μ)² – 2Ωμ Ω³ + μ³ = (Ω + μ)³ – 3Ωμ(Ω + μ) *The discriminant ∆ of a quadratic equation (types of roots of a quadratic equation)* In the quadratic formula –b±√b² – 4ac —––––––––– 2a The quantity b² – 4ac an the square root sign is known as the discriminant with symbol ∆.... the symbol is used to represent the discriminant where ∆ = b² – 4ac NB ✓If b² – 4ac = 0 the equation has equal roots or two identical real roots (repeated roots ) ✓ if b² – 4ac > 0 the equation has two real distinct roots ✓ if b² – 4ac < 0 the equation has imaginary or complex roots (no real roots) ✓if b² – 4ac ≥ 0 the equation has two real roots▪️▪️▪️▪️▪️▪️▪️ ✍️ SUCCESS SURE WITH OUR EXAMS RUNS THE BEST SINCE 2012 ✍️IN GOD WE TRUST AND EAGLEEYECEO AS THE HELPER TO UR SUCCESS ✍️YOUR SUCCESS IS MY PRIORITY 🙏 ✍️ KINDLY SAVE MY NUMBER 08023343546 AND TEXT ME IMMEDIATELY IN ORDER TO DO YOU REGISTRATION AND JOIN OUR ONLINE VIP GROUP MR EAGLEEYECEO THE BEST OF THE BEST

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CHEMISTRY TEST —✅ (a) (i) Explain briefly the fermentation process. (ii) Write a balanced equation for the fermentation of glucose. (iii) What substance must be added to glucose solution to ferment it? (iv) Explain briefly why tightly corked glass filled to the brim with palm wine shatters on standing. (b) State one industrial application of each of the following methods of separation: (i) crystallization; (ii) fractional distillation. (c ) Explain the following terms: (i) saponification; (ii) esterification. (d) Write a balanced equation to illustrate each of the terms in (c). (e) (i) What is hydrocarbon compound? (ii) Name two principal sources of hydrocarbons. Answers will be posted in the VIP

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EXAMINATION Date for 2022 EXAMS Below✍️💯: ▪WAEC-2022 9th MAY - 20th JUNE ▪NECO-2022 12th JULY to 22nd AUGUST .▪JAMB-2022 6th MAY to 30th MAY ▪IJMB-2022 15th JUNE to 30th JUNE . ▪JUPEB -2022 7th SEPTEMBER to 18th SEPTEMBER . ▪NABTEB-2022 20th JUNE to 19th JULY

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