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Number & Algebra

Algebraic Processes: Quadratic Equations, Variation, Sequences & Series, and Inequalities

SSS 1 · SSS 2 · SSS 3

  • ✓By the end of this lesson students will be able to solve quadratic equations using various methods.
  • ✓By the end of this lesson students will be able to apply the concepts of direct, inverse, joint, and partial variation to solve problems.
  • ✓By the end of this lesson students will be able to determine the nth term and sum of terms for arithmetic and geometric progressions.
  • ✓By the end of this lesson students will be able to solve linear and quadratic inequalities and represent their solutions on a number line.

Key concepts

Quadratic Equations

A quadratic equation is a polynomial equation of the second degree, meaning it contains at least one term in which the unknown variable is squared, but no term with a higher power. The general form is ax² + bx + c = 0, where 'x' is the variable, and 'a', 'b', and 'c' are real numbers with 'a' not equal to zero. Solutions to quadratic equations are also known as roots.

x = [-b ± √(b² - 4ac)] / 2a (General Formula)
Variation

Variation describes how one quantity depends on another. There are different types of variation:

Direct Variation

When a quantity 'y' varies directly as another quantity 'x', it means that 'y' is proportional to 'x'. As 'x' increases, 'y' increases proportionally, and vice-versa. This relationship can be written as y ∝ x, which translates to y = kx, where 'k' is the constant of proportionality.

y = kx
Inverse Variation

When a quantity 'y' varies inversely as another quantity 'x', it means that 'y' is proportional to the reciprocal of 'x'. As 'x' increases, 'y' decreases proportionally, and vice-versa. This relationship can be written as y ∝ 1/x, which translates to y = k/x, where 'k' is the constant of proportionality.

y = k/x
Joint Variation

Joint variation occurs when a quantity 'y' varies directly as the product of two or more other quantities, say 'x' and 'z'. This relationship can be written as y ∝ xz, which translates to y = kxz, where 'k' is the constant of proportionality.

y = kxz
Partial Variation

Partial variation occurs when a quantity 'y' is partly constant and partly varies directly or inversely as another quantity 'x'. For example, if 'y' is partly constant and partly varies directly as 'x', the relationship is y = kx + c, where 'k' and 'c' are constants.

y = kx + c (for direct partial variation)
Sequences and Series

A sequence is an ordered list of numbers (terms), while a series is the sum of the terms of a sequence. The two main types of sequences and series studied are Arithmetic Progression (AP) and Geometric Progression (GP).

Arithmetic Progression (AP)

An arithmetic progression is a sequence of numbers such that the difference between consecutive terms is constant. This constant difference is called the common difference (d).

nth term: T_n = a + (n-1)d\nSum of n terms: S_n = n/2 [2a + (n-1)d] or S_n = n/2 (a + T_n)
Geometric Progression (GP)

A geometric progression is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (r).

nth term: T_n = ar^(n-1)\nSum of n terms: S_n = a(r^n - 1)/(r - 1) for r > 1 or S_n = a(1 - r^n)/(1 - r) for r < 1\nSum to infinity: S_∞ = a/(1 - r) for |r| < 1
Inequalities

An inequality is a mathematical statement that compares two expressions using inequality signs: < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). Solving an inequality means finding the range of values for the variable that makes the statement true.

Linear Inequalities

These are inequalities involving expressions of degree one. They are solved similarly to linear equations, with the key difference being that multiplying or dividing by a negative number reverses the inequality sign.

Quadratic Inequalities

These are inequalities involving expressions of degree two. They are typically solved by finding the roots of the corresponding quadratic equation, then testing intervals or sketching the parabola to determine where the inequality holds true.

Key facts to remember

  • 1The general form of a quadratic equation is ax² + bx + c = 0, where a ≠ 0.
  • 2The general formula x = [-b ± √(b² - 4ac)] / 2a provides the roots of any quadratic equation.
  • 3In variation problems, 'k' is the constant of proportionality and must be determined first.
  • 4For an Arithmetic Progression (AP), the common difference 'd' is constant, and T_n = a + (n-1)d.
  • 5For a Geometric Progression (GP), the common ratio 'r' is constant, and T_n = ar^(n-1).
  • 6When multiplying or dividing both sides of an inequality by a negative number, the inequality sign must be reversed.
  • 7Solutions to inequalities are typically ranges of values, often represented on a number line.
  • 8For quadratic inequalities, critical points are found by solving the corresponding quadratic equation.

Worked examples

Example 1

Solve the quadratic equation 3x² - 7x + 2 = 0 using the general formula.

IIdentify the coefficients: For ax² + bx + c = 0, we have a = 3, b = -7, c = 2.
IIState the general formula: x = [-b ± √(b² - 4ac)] / 2a
IIISubstitute the values into the formula: x = [-(-7) ± √((-7)² - 4(3)(2))] / (2(3))
IVSimplify the expression: x = [7 ± √(49 - 24)] / 6
VContinue simplifying: x = [7 ± √25] / 6
VICalculate the square root: x = [7 ± 5] / 6
VIIFind the two possible values for x:
VIIIx₁ = (7 + 5) / 6 = 12 / 6 = 2
9x₂ = (7 - 5) / 6 = 2 / 6 = 1/3

Answer

x = 2 or x = 1/3

The general formula is always applicable, even when factorisation is difficult or impossible.

Example 2

The cost, C, of producing a certain item is partly constant and partly varies directly as the number of items, n, produced. If the cost is N1500 for 10 items and N2200 for 20 items, find the cost of producing 35 items.

IFormulate the variation equation: Since C is partly constant and partly varies directly as n, we write C = k + mn, where 'k' and 'm' are constants.
IIUse the given information to form simultaneous equations:
IIIWhen n = 10, C = 1500: 1500 = k + 10m (Equation 1)
IVWhen n = 20, C = 2200: 2200 = k + 20m (Equation 2)
VSubtract Equation 1 from Equation 2 to eliminate 'k':
VI(2200 - 1500) = (k - k) + (20m - 10m)
VII700 = 10m
VIIISolve for 'm': m = 700 / 10 = 70
9Substitute 'm = 70' into Equation 1 to find 'k':
101500 = k + 10(70)
111500 = k + 700
12k = 1500 - 700 = 800
13Write the complete variation equation: C = 800 + 70n
14Find the cost for 35 items by substituting n = 35:
15C = 800 + 70(35)
16C = 800 + 2450
17C = 3250

Answer

The cost of producing 35 items is N3250.

Always clearly define your constants in partial variation problems.

Example 3

Solve the inequality x² - x - 12 < 0 and represent the solution on a number line.

IConvert the inequality to an equation to find the critical points: x² - x - 12 = 0
IIFactorise the quadratic equation: (x - 4)(x + 3) = 0
IIIFind the roots (critical points): x - 4 = 0 => x = 4; x + 3 = 0 => x = -3
IVThese critical points divide the number line into three intervals: x < -3, -3 < x < 4, and x > 4.
VTest a value from each interval in the original inequality x² - x - 12 < 0:
VIInterval 1 (x < -3): Let x = -4. (-4)² - (-4) - 12 = 16 + 4 - 12 = 8. 8 < 0 is False.
VIIInterval 2 (-3 < x < 4): Let x = 0. (0)² - (0) - 12 = -12. -12 < 0 is True.
VIIIInterval 3 (x > 4): Let x = 5. (5)² - (5) - 12 = 25 - 5 - 12 = 8. 8 < 0 is False.
9The inequality x² - x - 12 < 0 is true for the interval -3 < x < 4.
10Represent the solution on a number line: Draw a number line, mark -3 and 4 with open circles (since the inequality is strictly less than, not less than or equal to), and shade the region between -3 and 4.

Answer

-3 < x < 4

For quadratic inequalities, sketching the parabola y = x² - x - 12 can also quickly show where the graph is below the x-axis (y < 0).

Common mistakes

  • ✗Incorrectly applying the signs in the general quadratic formula, especially with negative 'b' values.
  • ✗Forgetting to include the constant of proportionality 'k' in variation equations, or mixing up direct and inverse relationships.
  • ✗Confusing the formulae for Arithmetic Progression (AP) and Geometric Progression (GP).
  • ✗Failing to reverse the inequality sign when multiplying or dividing by a negative number.
  • ✗Incorrectly representing inequality solutions on a number line (e.g., using closed circles for strict inequalities or vice-versa).

Exam tips

  • ★Always show all your working steps clearly and logically, as marks are often awarded for method.
  • ★Memorise all relevant formulae for quadratic equations, variation, and sequences/series.
  • ★For variation problems, carefully read the question to correctly identify the type of variation (direct, inverse, joint, partial).
  • ★After solving equations or inequalities, substitute your answer(s) back into the original problem to verify your solution.

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