1. QUADRATICS P1
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SURDS
RATIONALIZING THE DENOMINATOR
INDICES & EXPONENTIAL
REDUCIBLE
COMPLETING THE SQUARE
INEQUALITIES
NATURE OF ROOTS (DISCRIMINANT)
SIMULTANEOUS EQUATION
SURDS
25 = 5
121 = 11 Rational number
8! = 2
5
17! Surds, Irrational number ( ๐, โฎ )
32
Rules:
๐ ร ๐ = ๐๐ ๐.๐. 3 ร 2 = 6
๐ รท ๐ = ๐๐ ๐.๐. 8 รท 6 =
86
1๐=
๐๐ ๐.๐.
32=
32ร
22=3 22
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Examples:
a. 54 b. 98 c. 32 โ 18 d. 75 + 48 e. 32ร 15 รท 24 f. 112 โ 63 โ 28 g. !
!
Rationalize
a. !! d. !! !
!!! !
b. !!! !
c. !! !!! !
Indices
1. am x an = am + n e.g. 75 x 73 = 78 2. am รท an = am โ n e.g. 96 รท 92 = 94 3. (am)n = amn e.g. (23)4 = 212 4. a-m = !
!! e.g. 4-3 = !
!!= !
!"
5. ๐!! = ๐!! = ( ๐! )! e.g. 8
!! = ( 8! )! = 2! = 16
6. a0 = 1
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Examples: Simplify
a. 22 x 24 c. !!!
!!!
b. !"ร!!
!!ร! d. 3x3y ร4๐ฅ๐ฆ!
Examples: Evaluate
a. 16!!!
b. 144! !
c. 27!!
d. (!!)!!
Examples: Solve the following equation
a. 2x = 32 b. 2x = 0.5 c. 24 x 27 = 2x d. 78 รท 7x = 49
e. ๐ฅ!!! = 8
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f. 8! = !!"#
Exponential Equations
If u = 2x then
a. 22x b. 2x + 2 c. 2-ยญโx d. 4x e. 16-ยญโx
Reducible to quadratic equations Solving exponential equations
a. 22x โ 2x โ 2 = 0 c. 16x โ 5(22x-ยญโ1) + 1 = 0
b. 3x+1 + 32-ยญโx โ 28 = 0
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1. x4 โ 5x2 + 4 = 0 4) 2 ๐ฅ = 8 โ ๐ฅ
2. x4 โ 9x2 โ 10 = 0
3. x6 โ 9x3 + 8 = 0 5) t -ยญโ 5 ๐ก โ 14 = 0
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Completing the square
ax2 + bx + c = 0
To solve quadratic equation using completing the square method:
a[(๐ฅ + !!!)! โ ( !
!!)!] + ๐
Examples:
a. x2 โ 4x d. 2x2 + 6x โ 1
b. x2 + 4x + 6 e. 3x2 โ x -ยญโ 7
c. x2 โ 6x โ 3 f. 5 โ 3x โ x2
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Solving quadratic equations (Find x) Using completing the square to solve quadratic equations (to find value of x)
Examples:
a. 3x2 โ 6x โ 1 = 0
b. x2 + 4x โ 7 = 0
c. 2x2 โ 5x โ 1 = 0
Maximum/ minimum values y
f(x) = 2x2 โ 4x โ 1
= 2[(x โ 1)2 โ 1 -ยญโ !! ]
= 2[(x โ 1)2 -ยญโ !! ] x
= 2(x โ 1)2 -ยญโ 3 (1,-ยญโ3)
y = -ยญโ3 (minimum value) stationary/turning point x = 1
minimum value = -ยญโ3 when x = 1. Range is f(x) โฅ โ3
1. Given that f(x) = 5 โ 3x โ x2. Find the minimum value and the value of x.
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2. Find the stationary point of the following quadratic equations
a) y = x2 + 4x
b) y = ( x + 4) (3 โ x)
Quadratic Inequalities 1. x2 + x โ 2 < 0
(x + 2) ( x โ 1) < 0 x = -ยญโ2 and 1
-ยญโ2 1
โด โ2 < ๐ฅ < 1
-ยญโ2 < x < 1
2. x2 + 3x + 1 โฅ 0
3. -ยญโ2 + 3x โ x2 < 0
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Nature of Roots (x)
x = !!ยฑ !!!!!"!!
when you use this formula, 3 results can happen:
1. b2 โ 4ac > 0 and you will get two different (distinct) answers for x or two distinct real roots.
2. b2 โ 4ac = 0 and you will get the repeated root (i.e. same answer twice). e.g. x2 โ 6x + 9 = 0 b2 โ 4ac = (-ยญโ6)2 โ 4(1)(9) = 0 (x โ 3) (x โ 3) = 0 x = 3
3. b2 โ 4ac < 0 then there are No Real Roots to the equation e.g. x2 + x + 2 = 0
b2 โ 4ac = (1)2 โ 4 (1)(2) = -ยญโ7 solve using quadratic formula: !!ยฑ !!!
not valid
The answers to equations are called the Roots and b2 โ 4ac is called the discriminant
b2 โ 4ac > 0 is Two Real Distinct Roots
b2 โ 4ac = 0 is Equal/same Roots (Repeated roots)
b2 โ 4ac < 0 is No Real Root (Complex root)
b2 โ 4ac > 0 b2 โ 4ac = 0 b2 โ 4ac < 0 (Intersect) 2 distinct real roots (Tangent to the curve) (Does not intersect) Equal root/same root No real root/lies above x-ยญโaxis
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Examples: A. Use the discriminant to determine the nature of the roots of the following equations 1. x2 โ 6x + 9 = 0 2. 2x2 โ 5x + 3 = 0 3. 1 2x
2 + 1 3 ๐ฅ +14= 0
B. The equation x(x โ 2) + k2 = k(2x โ 1) is satisfied by two distinct real values of x. Find the range of values of k.
C. Find the set of values of k for which the line y = kx โ 4 intersects the curve y = x2 โ 2x at two distinct points.
D. The equation of a curve is y = x2 โ 3x + 4. Show that the whole of the curve lies above
the x-ยญโaxis.
E. The equation of a curve is xy = 12 and the equation of a line l is 2x + y = k, where k is a constant. Find the set of values of k for which l does not intersect the curve.
F. Find the set of values of k for which the equation 2x2 โ 10x + 8 = kx has no real roots.
G. Find the value of constant c for which the line y = 2x + c is a tangent to the curve y2 = 4x.
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H. Find the value of p in each of the following equations given that they have EQUAL ROOTS. i) x2 + (p โ 1)x + 64 = 0 ii) 3x2 โ (p โ 4)x โ (2p + 1) = 0
Simultaneous Equations Involving Quadratics 1.
y = 3 โ x2 y = 7 โ 4x
2. y = 2x โ 7 2xy + 3y + 5x + 11 = 0
3. Find the point of intersection between the line y = 3x โ 1 and 2y โ 5x + 7 = 0 4. The equation of a curve xy = 12 and the equation of a line l is 2x + y = k, where k is a constant. In
the case where k = 11, find the coordinates of the points of intersection of l and the curve. 5. The curve y = 9 โ 6/x and the line y + x = 8 intersect at two points. Find the coordinates of the
two points.