CSEC/CXC Algebra

Example 1: Simplify 3x + 2y - x + 4y

Solution:

• Combine like terms: (3x - x) + (2y + 4y)
• Simplify: 2x + 6y

Example 2: Evaluate 2a² - 3b when a = 4 and b = 2

Solution:

• Substitute values: 2(4)² - 3(2)
• Calculate: 2(16) - 6 = 32 - 6 = 26

Example 3: Write an expression for "5 less than twice a number"

Solution:

• Let the number be x
• Twice the number: 2x
• 5 less: 2x - 5

Example 1: Factorize completely: 6x² + 11x + 4

Solution:

• Find two numbers whose product is 6×4=24 and sum is 11 → 8 and 3
• Rewrite: 6x² + 8x + 3x + 4
• Factor by grouping: 2x(3x + 4) + 1(3x + 4)
• Final factors: (2x + 1)(3x + 4)

Example 2: Factorize: x² - 9y²

Solution:

• Difference of squares: a² - b² = (a + b)(a - b)
• Factors: (x + 3y)(x - 3y)

Example 3: Factorize: 5x³ - 20x

Solution:

• Factor out GCF: 5x(x² - 4)
• Factor further: 5x(x + 2)(x - 2)

Example 1: Solve for x: 3(x - 4) = 2x + 5

Solution:

• Expand: 3x - 12 = 2x + 5
• Subtract 2x from both sides: x - 12 = 5
• Add 12 to both sides: x = 17

Example 2: Solve: 2(x + 3) - 5 = 3x - 1

Solution:

• Expand: 2x + 6 - 5 = 3x - 1 → 2x + 1 = 3x - 1
• Subtract 2x: 1 = x - 1
• Add 1: x = 2

Example 3: Solve: (x + 5)/2 = 3x - 2

Solution:

• Multiply both sides by 2: x + 5 = 6x - 4
• Subtract x: 5 = 5x - 4
• Add 4: 9 = 5x
• Divide by 5: x = 9/5 or 1.8

Example 1: Solve x² - 5x + 6 = 0

Solution:

• Factorize: (x - 2)(x - 3) = 0
• Set each factor equal to zero: x - 2 = 0 or x - 3 = 0
• Solutions: x = 2 or x = 3

Example 2: Solve 2x² + 5x - 3 = 0 using the quadratic formula

Solution:

• Quadratic formula: x = [-b ± √(b² - 4ac)]/(2a)
• a=2, b=5, c=-3
• Discriminant: (5)² - 4(2)(-3) = 25 + 24 = 49
• Solutions: x = [-5 ± √49]/4 → x = (-5 ± 7)/4
• x = (-5+7)/4 = 0.5 or x = (-5-7)/4 = -3

Example 3: Solve by completing the square: x² + 6x + 5 = 0

Solution:

• Move constant: x² + 6x = -5
• Complete square: (6/2)² = 9 → x² + 6x + 9 = -5 + 9
• Simplify: (x + 3)² = 4
• Square root both sides: x + 3 = ±2
• Solutions: x = -3+2 = -1 or x = -3-2 = -5

Example 1: Solve the system: 2x + y = 7 3x - y = 3

Solution:

• Add the two equations: 5x = 10 → x = 2
• Substitute x=2 into first equation: 2(2) + y = 7 → y = 3
• Solution: x = 2, y = 3

Example 2: Solve by substitution: y = 2x + 1 3x + 2y = 16

Solution:

• Substitute y into second equation: 3x + 2(2x + 1) = 16
• Expand: 3x + 4x + 2 = 16 → 7x + 2 = 16
• Solve: 7x = 14 → x = 2
• Find y: y = 2(2) + 1 = 5
• Solution: x = 2, y = 5

Example 3: Solve: 4x + 3y = 10 5x - 2y = 1

Solution:

• Multiply first by 2: 8x + 6y = 20
• Multiply second by 3: 15x - 6y = 3
• Add them: 23x = 23 → x = 1
• Substitute x=1 into first equation: 4(1) + 3y = 10 → y = 2
• Solution: x = 1, y = 2

Example 1: Plot the graph of y = 2x - 1

Solution:

• When x=0, y=-1 → (0,-1)
• When x=1, y=1 → (1,1)
• Plot these points and draw a straight line through them

Example 2: Find the equation of the line with gradient 3 passing through (1,4)

Solution:

• Use point-slope form: y - y₁ = m(x - x₁)
• Substitute: y - 4 = 3(x - 1)
• Simplify: y = 3x - 3 + 4 → y = 3x + 1

Example 3: Find the equation of the line through (2,5) and (4,11)

Solution:

• Gradient (m) = (11-5)/(4-2) = 6/2 = 3
• Using point (2,5): y - 5 = 3(x - 2)
• Simplify: y = 3x - 6 + 5 → y = 3x - 1

Example 1: Sketch the graph of y = x² - 4x + 3

Solution:

• Find roots: x² - 4x + 3 = 0 → (x-1)(x-3)=0 → x=1, x=3
• Find vertex: x-coordinate is midpoint of roots → x=2
• When x=2, y=(2)²-4(2)+3=-1 → vertex at (2,-1)
• Sketch parabola through these points

Example 2: Sketch y = -x² + 2x + 8

Solution:

• Find roots: -x² + 2x + 8 = 0 → x² - 2x - 8 = 0 → (x-4)(x+2)=0 → x=4, x=-2
• Find vertex: x-coordinate is midpoint → x=1
• When x=1, y=-(1)²+2(1)+8=9 → vertex at (1,9)
• Parabola opens downward

Example 3: Find the equation of the parabola with vertex (2,-3) passing through (4,5)

Solution:

• Vertex form: y = a(x - h)² + k where (h,k) is vertex
• Substitute vertex: y = a(x - 2)² - 3
• Use point (4,5): 5 = a(4-2)² - 3 → 5 = 4a - 3 → a = 2
• Equation: y = 2(x - 2)² - 3 or y = 2x² - 8x + 5