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CCEA MATHS AS NOTES

08/09/2022

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CURVE SKETCHING CONT.
Welch the curve 7-6x-x²
y =
x²+6x-7:0
(x+7)(x-1)=0
x=-7 x=1
(-7,0) (1,0)
-x² - 6x +7=0
- [(x+3) ²-9-7-0
-(x+3)² +16=0

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CURVE SKETCHING CONT.
Welch the curve 7-6x-x²
y =
x²+6x-7:0
(x+7)(x-1)=0
x=-7 x=1
(-7,0) (1,0)
-x² - 6x +7=0
- [(x+3) ²-9-7-0
-(x+3)² +16=0

Register

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CURVE SKETCHING CONT.
Welch the curve 7-6x-x²
y =
x²+6x-7:0
(x+7)(x-1)=0
x=-7 x=1
(-7,0) (1,0)
-x² - 6x +7=0
- [(x+3) ²-9-7-0
-(x+3)² +16=0

Register

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By signing up you accept Terms of Service and Privacy Policy

CURVE SKETCHING CONT. Welch the curve 7-6x-x² y = x²+6x-7:0 (x+7)(x-1)=0 x=-7 x=1 (-7,0) (1,0) -x² - 6x +7=0 - [(x+3) ²-9-7-0 -(x+3)² +16=0 MAX@(-3,16) (-3,16) Ex2D x=0; y = 7 (0₁7) THE DISCRIMINANT a=3 b=-4 C=2 When-b²-4ac70 TWO REAL ROOTS b²-4ac-O ONE REPEATED BOUR → b²-4ac40 No REAL ROOTS 6²-4ac is the discriminant of ax²+bx+c Find the number of distinct roots in → 3x² - 4x +2=0 ax²+bx+c=0 b²-4ac (-4)²-4(3)(2) 16-24-8 LO no real roots * x² +6x-1=0 6²-4ac = (6) ² - 4(1)(-1) 36+4 a= 1 b=6 C=-1 = - 4070 = 2 real roots repeated roots Find the values of k for which 2(4-1) x² - 2kx + k-1=0 has equal roots. a a= b= 2k c- k-1 2(k-1) =4 b²-4ac=0 (24)= 4(2(4-1))(k-1)=0 4628 (4-1) 14-1) = 0 442-8 (1²-2k+1) O 442-842-161-8-0 -46²-16k-8-0 -k²-4k-2=0 2²+4+k+2=0 (6-2)²-2₁+2=0 (4-2)² = 2 k=2¹/√/2 Write 2x²-8x + 7 in the form a (x+b)² + c complete the square 2[x²-4x]+7 2 [(x-2)²-4]+7 2(x-2)²-1 Write clown the minimum value of f(x) = 2x²-8x+7 and state the minimum value of xe where it occurs. 2(x-2)²-1 MIN = -1 when x=2 dolve the equation 2x² - 8x+7=0 leaving your answer in surd form. 2(x-2)²-1=0 2 (x-2)² = 1 (x-2)² = 11/12 x-2-√√ x= 2√√ PURE MATHS INDICES M a x a = a am aº a (a™)" = a' = ♥ eg. a SOLVING QUADRATICS Solve: x² +22x+121=0 (m+n) (m-n) = x² +11ac +11ac +12) = 0 x (x+11) + \\ ( x + 1) = (x+11)(x+11)=0 (x+11)² = 0 x=-11 repeated root +$ 22 = a eg. Solve: 70²-63=0, r²-9-0 (r+3)(r-3) = 0 r: -3:3 COMPLETING THE SOVARE - (x + ²)² = (x+¹=2)(x+²1/12) > 11+11:22 121 + 11×11 = 121 : €x 10+1= x=-6²b²-4ac 2a Je2 +20+ वे fill in the missing value to complete the square. x²-3x+ x²-3x+ SUROS eg. sumplify √45 + √80² √9x5...

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Alternative transcript:

+√16x5 3√5 +4J5 = 755 eg. expand & simplify: (√F-2)(√7+2) (√7-2)(√√7+2) (√ā -√b)(√ä +√6)-a-b 45 3 x√7 3√7 3√ LVT VT VT 307 35 4√7 4√49 4x7 28 eg simplify Ex 28+2 Ex IF 2./3-1 eg. simplify √5 +1 2√√√3-1 x√3-1 √3 +1 *√√3-1 • x² + 3x +2.25 complete the aquare for the following • ac² +bac →> (oc+ 3) ³² - 9 *XC²+Fxc-11→ (x - 3²) ²-409-10-11 = (x - ²1/1) ² - 93 *3ac² +6x-5 =3(x²+2x)-5 3+1)²-1]-5= 3(x+1) ²-3-5 3(x+1)²-8 √49 +2√7-2√7-4 = 7-4 = 3 Perfect squares 4x20 16x5 : (2√3-1)(√3-1) (√3+1)(√3-1) = 2(3)-2√3-√3+1 3•√5+√5-1 7-3√3 2 eg. 5x²+4x+1 5(3²+)+\ 5 [(x + ²)²-²]+1 5(x+²+1 25 5(x + ²)²³² +5 Ex 20 COMPLETING THE SQ. CONT. x² + 7x0-11=0 (ac+)²-49-11-0 4१ 93 4 (+)²- (oc+ 1)² = 93 SQUARE ROOT BOTH SIDES x+=√√√ X=- - eg. THIS CAN ALSO BE USED TO HELP FIND MAXIMUMOR MINIMUM EXPRESSIONS. find the min values: x² - 6x + 6 (oc-3)*-9+6 (x-3)²-3 minimum value=-3 x = 3 >X² + 7x₁ + 7 (ac+ 1)²-42+7 (+)²-21 munum um valve eg. 24444 only 3.5 both Jides if = 0 Man value = 9 x=-1 +√95 a x=-2 -32 find the man values: 8-2x-x² -(x²+2x-8) -[(x+1)=-1-8] -(x+1)²+9 → 12+x-x² -(x²-x-12) 49 4 - [(x-1)== -12] -(x-1)²+ Max valve = 44 x = 1/2 49 * SKETCHING A FUNCTION +ve - ve roots of the function- - the maximum / minimum value of y Metch the graph: y=x²-7xc-4 (x-1) ²49-2-0 4 (x-7) ²65 =O x-1=√√65 4.55 65 (√2)(√60) y=-4 (0₁-4) X=O y= 0²-7(0)-4 MIN@ (x-7) ²-63 65 (3-65) *6 X 2 Hir where they cut the 2 HN +√65