Quadratic Equations

Core Concepts

A quadratic equation is a second-degree polynomial equation. Its standard form is: $$ax^2 + bx + c = 0$$ where $x$ is the unknown variable, and $a$, $b$, and $c$ are constants with $a \neq 0$.

1. Solving by Factorization This method relies on the Zero Product Property: if $A \times B = 0$, then either $A = 0$, $B = 0$, or both. To factorize a quadratic expression $ax^2 + bx + c$:

Find two numbers whose product is $a \times c$ and whose sum is $b$.
Split the middle term ($bx$) using these two numbers.
Factor by grouping to write the equation in the form $(px + q)(rx + s) = 0$.
Equate each factor to zero to find the roots.

2. Completing the Square This method involves manipulating the equation to form a perfect square trinomial on one side:

Start with $ax^2 + bx + c = 0$.
Divide the entire equation by $a$ so the coefficient of $x^2$ is 1: $x^2 + \frac{b}{a}x + \frac{c}{a} = 0$.
Move the constant term to the right side: $x^2 + \frac{b}{a}x = -\frac{c}{a}$.
Add the square of half the coefficient of $x$ to both sides: $\left(\frac{b}{2a}\right)^2$.
Factor the left side as a perfect square: $\left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2}$.
Take the square root of both sides (using $\pm$) and solve for $x$.

3. General Solution (The Quadratic Formula) By completing the square on the general equation, we derive the quadratic formula. This formula can reliably solve any quadratic equation: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ The expression under the radical, $b^2 - 4ac$, is called the discriminant. It dictates the nature of the roots:

If $b^2 - 4ac > 0$, there are two distinct real roots.
If $b^2 - 4ac = 0$, there is exactly one repeated real root.
If $b^2 - 4ac < 0$, there are no real roots.

4. Equations Reducible to Quadratic Form Some higher-degree or fractional equations can be transformed into a quadratic equation via a suitable substitution. For example, an equation like $\frac{a}{t^4} + \frac{b}{t^2} + c = 0$ can be simplified by letting $x = \frac{1}{t^2}$, which turns it into the standard quadratic form $ax^2 + bx + c = 0$.

Worked Examples

Example 1: Solving by Factorization Factorise the quadratic expression $3x^2 - 11x - 20$ by splitting the middle term, and hence solve $3x^2 - 11x - 20 = 0$.

Solution: We need two numbers that multiply to $a \times c = 3 \times (-20) = -60$ and add up to $b = -11$. By inspecting the factors of $-60$, we find that $-15$ and $4$ meet these criteria. Split the middle term: $$3x^2 - 15x + 4x - 20 = 0$$ Group the terms in pairs and factor out the greatest common factor (GCF) from each pair: $$3x(x - 5) + 4(x - 5) = 0$$ Factor out the common binomial $(x - 5)$: $$(x - 5)(3x + 4) = 0$$ Apply the Zero Product Property: $$x - 5 = 0 \quad \text{or} \quad 3x + 4 = 0$$ $$x = 5 \quad \text{or} \quad x = -\frac{4}{3}$$

Example 2: Formulating and Solving using the Quadratic Formula Rachel is three years older than her brother John. Three years to come, the product of their ages will be 130 years. Formulate a quadratic equation representing this information. Hence, by using the quadratic formula, find their present ages.

Solution: Let John's present age be $j$. Then Rachel's present age is $j + 3$.

In three years: John's age $= j + 3$ Rachel's age $= (j + 3) + 3 = j + 6$

The product of their ages in three years is 130: $$(j + 3)(j + 6) = 130$$ Expand the left side to formulate the quadratic equation: $$j^2 + 6j + 3j + 18 = 130$$ $$j^2 + 9j + 18 - 130 = 0$$ $$j^2 + 9j - 112 = 0$$

Now, solve using the quadratic formula where $a = 1$, $b = 9$, $c = -112$: $$j = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ $$j = \frac{-9 \pm \sqrt{9^2 - 4(1)(-112)}}{2(1)}$$ $$j = \frac{-9 \pm \sqrt{81 + 448}}{2}$$ $$j = \frac{-9 \pm \sqrt{529}}{2}$$ $$j = \frac{-9 \pm 23}{2}$$

This gives two possible values for $j$: $$j = \frac{14}{2} = 7 \quad \text{or} \quad j = \frac{-32}{2} = -16$$ Since age cannot be negative, we reject $-16$. Therefore, John's present age is $7$ years. Rachel's present age is $7 + 3 = 10$ years.

NECTA Exam Focus

In the NECTA CSEE Basic Mathematics syllabus, Quadratic Equations are a high-frequency topic, almost always appearing in Paper 1. The questions are designed to test both algebraic manipulation and the ability to model real-world situations mathematically.

Recurring Themes:

Direct Factorization: Questions frequently ask students to factorize algebraic expressions by "splitting the middle term" explicitly, testing their foundational algebra skills.
The Quadratic Formula: Applying the general formula is a core requirement, often specified directly in the prompt. Students must memorize the formula and be able to evaluate discriminants accurately.
Word Problems: Age puzzles, geometric dimensions (area/perimeter), and numbers that require formulating a quadratic equation from a textual description are standard.
Substitution Techniques: Transforming equations that are not strictly quadratic (e.g., rational equations or higher-degree polynomials) into quadratic form using a substitution like $x = \frac{1}{t^2}$ is commonly tested in the advanced parts of the examination.

Common Pitfalls:

Sign Errors: Mistakes are highly prevalent when determining the factors to split the middle term or when substituting negative values for $b$ and $c$ into the quadratic formula.
Neglecting the $\pm$: Forgetting that taking a square root in the quadratic formula yields both a positive and a negative path.
Ignoring Contextual Constraints: In word problems, students often forget to reject negative mathematical roots that have no physical meaning (like negative ages or lengths).
Incomplete Factorization: Leaving an expression partially factored instead of breaking it down fully to its linear factors.

Practice Problems

Use factorization method to solve the quadratic equation $x^2 - 9x + 14 = 0$.
If $x = \frac{1}{t^2}$, express the equation $\frac{2}{t^4} - \frac{3}{t^2} + 1 = 0$ in the quadratic form $ax^2 + bx + c = 0$.
Solve for $x$ in the quadratic equation obtained in the previous question.

Subtopics

Solving equations by factorization and completing the square
General solution of a quadratic equation

Crosswalk Notes

Cross-version relationships are drafted in data/curricula/crosswalks/csee-basic-mathematics-2005-to-mathematics-2023.json. Partial and 2005-only mappings remain reviewable.