Quadratic expressions and equations

Overview

A quadratic expression is an algebraic expression whose highest power of the variable is $2$. A quadratic equation is formed when such an expression is set equal to zero or to another expression.

This topic matters because quadratic thinking appears in algebra, graphs, geometry, area problems, substitution problems, and later function work. A learner who can factorise and solve quadratics can recognise turning points, roots, and useful equation forms.

Prerequisites

• Algebraic expressions and equations - Quadratics use algebraic terms, coefficients, brackets, and equation-solving steps.
• Exponents - The term $x^2$ is a power, and substitution problems may involve powers such as $t^{-2}$.
• Radicals - Square roots appear in the quadratic formula and in equations solved by taking square roots.
• Linear simultaneous equations - Linear solving skills help when a quadratic factor gives a simple linear equation.
• Coordinate geometry: gradient and straight-line equations - Graph features are easier when learners already understand axes and intercepts.

Learning Scope

This chapter covers recognising quadratic expressions, writing standard form, expanding brackets, factorising simple quadratics, solving quadratic equations by factorisation, square roots, completing the square, and the quadratic formula. It also introduces the graph of a quadratic function as a parabola, including roots, intercepts, and the vertex.

This page does not fully teach general functions, advanced curve sketching, or optimisation. It gives the quadratic ideas needed for Form II algebra and prepares learners for later work in Relations and functions, Graphs of relations and functions, and coordinate geometry.

Subtopics

Recognising Quadratic Expressions

A quadratic expression in $x$ has highest power $2$. The general form is:

$$ ax^2 + bx + c, \quad a \ne 0 $$

Here $a$, $b$, and $c$ are constants. The condition $a \ne 0$ is important because if $a=0$, the expression becomes linear, not quadratic.

Key insight: The highest power decides the type. The expression $4x^2-7x+3$ is quadratic, but $4x-7$ is linear and $4x^3-7x+3$ is cubic.

Standard Form And Coefficients

A quadratic equation is often written in standard form:

$$ ax^2 + bx + c = 0, \quad a \ne 0 $$

For example, in:

$$ 2x^2 - 5x - 3 = 0 $$

the coefficients are:

$$ a=2, \quad b=-5, \quad c=-3 $$

Key insight: Keep the signs with the coefficients. In $2x^2-5x-3$, the value of $b$ is $-5$, not $5$.

Expanding Products Of Linear Factors

Quadratic expressions often come from multiplying two linear factors:

$$ (x+p)(x+q) $$

Expand by multiplying each term in the first bracket by each term in the second bracket:

$$ \begin{aligned} (x+p)(x+q) &= x^2 + qx + px + pq \\ &= x^2 + (p+q)x + pq \end{aligned} $$

Example:

$$ \begin{aligned} (x+3)(x-5) &= x^2 - 5x + 3x - 15 \\ &= x^2 - 2x - 15 \end{aligned} $$

Factorising Simple Quadratics

Factorising reverses expansion. For a monic quadratic such as:

$$ x^2 + bx + c $$

look for two numbers whose sum is $b$ and whose product is $c$.

Example:

$$ x^2 + 7x + 12 $$

The two numbers are $3$ and $4$ because:

$$ 3+4=7, \quad 3 \times 4 = 12 $$

So:

$$ x^2 + 7x + 12 = (x+3)(x+4) $$

Key insight: Factorising is easier when learners check both the sum and product. One condition alone is not enough.

Factorising When The Leading Coefficient Is Not One

For a quadratic such as:

$$ ax^2+bx+c $$

where $a \ne 1$, one useful method is to split the middle term.

Example:

$$ 2x^2+7x+3 $$

Multiply $a$ and $c$:

$$ 2 \times 3 = 6 $$

Find two numbers whose product is $6$ and whose sum is $7$: they are $6$ and $1$. Split $7x$:

$$ \begin{aligned} 2x^2+7x+3 &=2x^2+6x+x+3 \\ &=2x(x+3)+1(x+3) \\ &=(2x+1)(x+3) \end{aligned} $$

Solving By Factorisation

If a product is zero, at least one factor must be zero:

$$ AB=0 \quad \Rightarrow \quad A=0 \text{ or } B=0 $$

Example:

$$ x^2 - 5x + 6 = 0 $$

Factorise:

$$ x^2 - 5x + 6 = (x-2)(x-3) $$

Then:

$$ (x-2)(x-3)=0 $$

So:

$$ x-2=0 \quad \text{or} \quad x-3=0 $$

Therefore:

$$ x=2 \quad \text{or} \quad x=3 $$

Key insight: The roots are the values of $x$ that make the quadratic equal to zero.

Solving By Taking Square Roots

Some quadratic equations have no $x$ term after rearranging:

$$ x^2 = k $$

If $k \ge 0$, then:

$$ x = \pm \sqrt{k} $$

Example:

$$ 3x^2 - 12 = 0 $$

Then:

$$ \begin{aligned} 3x^2 &= 12 \\ x^2 &= 4 \\ x &= \pm 2 \end{aligned} $$

Key insight: Solving $x^2=4$ gives two solutions, $x=2$ and $x=-2$. This is different from the principal square root $\sqrt{4}=2$.

Completing The Square

Completing the square rewrites a quadratic so that a squared bracket appears. For $x^2+bx+c$, take half the coefficient of $x$, square it, then adjust the constant.

Example:

$$ x^2 + 6x + 5 $$

Half of $6$ is $3$, and $3^2=9$. So:

$$ \begin{aligned} x^2 + 6x + 5 &= (x+3)^2 - 9 + 5 \\ &= (x+3)^2 - 4 \end{aligned} $$

This form can help solve equations and find graph features.

The Quadratic Formula

Any quadratic equation:

$$ ax^2+bx+c=0, \quad a \ne 0 $$

can be solved using:

$$ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} $$

The expression under the square root, $b^2-4ac$, helps show the nature of the roots:

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

Key insight: The formula works even when factorisation is not obvious, but signs must be handled carefully.

Graph Features Of A Quadratic

The graph of:

$$ y=ax^2+bx+c $$

is a parabola. If $a>0$, the parabola opens upward. If $a<0$, it opens downward.

Important features include:

• Roots or $x$-intercepts: where $y=0$.
• $y$-intercept: where $x=0$, usually the point $(0,c)$.
• Vertex: the turning point of the parabola.
• Axis of symmetry: the vertical line through the vertex.

For a future interactive layer, a web widget could let learners drag the coefficients $a$, $b$, and $c$ and watch how the roots, vertex, intercept, and opening direction change.

Substitution Into Quadratic Form

Some equations are not written as quadratics at first, but become quadratics after substitution.

For example, if:

$$ u=\frac{1}{t^2} $$

then:

$$ \frac{1}{t^4}=u^2 $$

So an equation involving $\frac{1}{t^4}$ and $\frac{1}{t^2}$ can be rewritten in the form:

$$ au^2+bu+c=0 $$

Key insight: Choose a substitution that turns repeated powers into $u^2$ and $u$.

Key Terms

• Quadratic expression: An expression whose highest variable power is $2$, such as $3x^2-2x+5$.
• Quadratic equation: An equation that can be written as $ax^2+bx+c=0$, where $a \ne 0$.
• Standard form: The arrangement $ax^2+bx+c=0$ for equations or $ax^2+bx+c$ for expressions.
• Coefficient: A number multiplying a variable term. In $-4x^2+7x-1$, the coefficient of $x^2$ is $-4$.
• Constant term: The term without a variable. In $x^2+5x-6$, the constant term is $-6$.
• Factor: A bracket or expression that multiplies with another to make the original expression.
• Root: A value of the variable that makes the equation equal to zero.
• Parabola: The curved graph of a quadratic function.
• Vertex: The turning point of a parabola.
• Axis of symmetry: The vertical line that divides a parabola into two matching halves.
• Discriminant: The expression $b^2-4ac$ in the quadratic formula.

Worked Examples

Example 1: Recognise Coefficients

Write $5x-2x^2+9=0$ in standard form and state $a$, $b$, and $c$.

Arrange terms from highest power to constant:

$$ -2x^2+5x+9=0 $$

Therefore:

$$ a=-2, \quad b=5, \quad c=9 $$

Final answer: standard form is $-2x^2+5x+9=0$.

Example 2: Factorise A Monic Quadratic

Factorise:

$$ x^2 - x - 12 $$

Find two numbers whose sum is $-1$ and whose product is $-12$. The numbers are $3$ and $-4$.

So:

$$ x^2 - x - 12 = (x+3)(x-4) $$

Check by expansion:

$$ (x+3)(x-4)=x^2-x-12 $$

Example 3: Solve By Factorisation

Solve:

$$ x^2 + 2x - 15 = 0 $$

Factorise:

$$ x^2 + 2x - 15 = (x+5)(x-3) $$

Then:

$$ \begin{aligned} (x+5)(x-3)&=0 \\ x+5&=0 \quad \text{or} \quad x-3=0 \end{aligned} $$

Therefore:

$$ x=-5 \quad \text{or} \quad x=3 $$

Example 4: Factorise When $a \ne 1$

Factorise:

$$ 3x^2 - 10x + 3 $$

Multiply $a$ and $c$:

$$ 3 \times 3 = 9 $$

Find two numbers with product $9$ and sum $-10$: they are $-9$ and $-1$. Split the middle term:

$$ \begin{aligned} 3x^2 - 10x + 3 &= 3x^2 - 9x - x + 3 \\ &= 3x(x-3) - 1(x-3) \\ &= (3x-1)(x-3) \end{aligned} $$

Final answer:

$$ (3x-1)(x-3) $$

Example 5: Use The Quadratic Formula

Solve:

$$ 2x^2 + 3x - 2 = 0 $$

Here:

$$ a=2, \quad b=3, \quad c=-2 $$

Use the formula:

$$ x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} $$

Substitute:

$$ \begin{aligned} x &= \frac{-3 \pm \sqrt{3^2 - 4(2)(-2)}}{2(2)} \\ &= \frac{-3 \pm \sqrt{9+16}}{4} \\ &= \frac{-3 \pm 5}{4} \end{aligned} $$

So:

$$ x=\frac{2}{4}=\frac{1}{2} $$

or:

$$ x=\frac{-8}{4}=-2 $$

Final answer:

$$ x=\frac{1}{2} \quad \text{or} \quad x=-2 $$

Example 6: Rewrite Into Quadratic Form

Let $u=\frac{1}{t^2}$. Express:

$$ \frac{3}{t^4}-\frac{8}{t^2}+4=0 $$

in quadratic form.

Since:

$$ u=\frac{1}{t^2} $$

then:

$$ u^2=\frac{1}{t^4} $$

Substitute:

$$ \begin{aligned} \frac{3}{t^4}-\frac{8}{t^2}+4&=0 \\ 3u^2-8u+4&=0 \end{aligned} $$

Final answer:

$$ 3u^2-8u+4=0 $$

Common Mistakes

• Mistake: Calling $7x+3=0$ a quadratic equation.

Correction: It is linear because the highest power of $x$ is $1$.

• Mistake: Forgetting that $a$ cannot be zero in $ax^2+bx+c=0$.

Correction: If $a=0$, the $x^2$ term disappears.

• Mistake: Losing negative signs when identifying coefficients.

Correction: In $x^2-8x+15$, $b=-8$.

• Mistake: Factorising using numbers with the correct product but wrong sum.

Correction: For $x^2+x-12$, use $4$ and $-3$, not $3$ and $-4$.

• Mistake: Stopping after factorising and not solving each factor.

Correction: From $(x-2)(x+7)=0$, write $x=2$ or $x=-7$.

• Mistake: Taking only the positive square root when solving $x^2=25$.

Correction: The solutions are $x=5$ and $x=-5$.

• Mistake: Writing the quadratic formula with the wrong denominator.

Correction: The denominator is $2a$, not only $2$.

• Mistake: Treating the $y$-intercept as the same thing as a root.

Correction: Roots are where $y=0$; the $y$-intercept is where $x=0$.

Practice Tasks

Direct Understanding

1. State whether $4x^2-9x+2$ is quadratic. Give a reason.
2. In $-3x^2+8x-6=0$, state $a$, $b$, and $c$.
3. Explain why $x^3+x^2+1$ is not a quadratic expression.
4. Name the graph shape of $y=x^2-4x+3$.

Skill Practice

1. Expand $(x+6)(x-2)$.
2. Expand $(2x-1)(x+5)$.
3. Factorise $x^2+9x+20$.
4. Factorise $x^2-4x-21$.
5. Factorise $2x^2+5x+2$.
6. Factorise $3x^2+x-2$.

Solving Quadratics

1. Solve $x^2-7x+10=0$ by factorisation.
2. Solve $x^2+3x-18=0$ by factorisation.
3. Solve $4x^2-25=0$ by taking square roots.
4. Solve $2x^2+x-6=0$.
5. Use the quadratic formula to solve $x^2-2x-8=0$.
6. Use the quadratic formula to solve $3x^2+2x-1=0$.

Graph And Interpretation

1. For $y=x^2-5x+6$, find the roots.
2. For $y=2x^2-3x+4$, state the $y$-intercept.
3. State whether $y=-x^2+4x+1$ opens upward or downward.
4. Complete the square for $x^2-8x+7$.

Application And Substitution

1. A rectangle has length $(x+4)$ cm and width $(x-1)$ cm. Write an expression for its area.
2. The area of a rectangle is $x^2+5x+6$. Factorise the expression to suggest possible side lengths.
3. If $u=x^2$, rewrite $2x^4-7x^2+3=0$ in terms of $u$.
4. If $p=\frac{1}{y}$, rewrite $\frac{3}{y^2}-\frac{5}{y}+2=0$ in quadratic form.

Edge Cases

1. Explain why $x^2+4=0$ has no real roots.
2. Find the discriminant of $x^2-6x+9=0$ and interpret it.
3. Give an example of a quadratic equation with roots $-2$ and $5$.
4. Explain why the expression $(x+3)^2$ has a repeated root when set equal to zero.

Generated Question Layer

• Conceptual questions: Ask learners to identify quadratic expressions, standard form, coefficients, constants, roots, factors, vertices, and intercepts.
• Expansion questions: Generate products of two linear brackets, including signs such as $(x-a)(x+b)$ and leading coefficients such as $(2x+1)(x-3)$.
• Factorisation questions: Start with monic quadratics, then include quadratics where $a \ne 1$.
• Solving questions: Generate equations solvable by factorisation, square roots, completing the square, and the quadratic formula.
• Graph questions: Ask learners to connect roots with $x$-intercepts, $c$ with the $y$-intercept, and the sign of $a$ with opening direction.
• Substitution questions: Generate equations that become quadratic after setting $u=x^2$, $u=\frac{1}{x}$, or $u=\frac{1}{t^2}$.
• Edge cases: Include repeated roots, no real roots, non-factorisable real-root examples, sign errors, and equations that are linear after simplification.

Learner Aid Opportunities

• chart: Show how $a$, $b$, and $c$ are read from standard form, including sign errors.
• diagram: Link a factorised quadratic to its roots on a number line.
• graph: Show roots, vertex, axis of symmetry, and $y$-intercept on a parabola.
• interactive: Let learners adjust $a$, $b$, and $c$ and observe changes in roots, vertex, intercept, and opening direction.
• LLM tutor: Coach factorisation by checking the required sum and product before solving.

Exam-Derived Signals

The 2021-2025 automatic topic-frequency file counts topic-quadratic-expressions-and-equations three times. These records are useful as assessment signals, but they remain unreviewed extraction data unless a maintainer checks them against the original papers.

The 2022 CSEE Basic Mathematics examination format groups Algebra/Quadratic equations as one listed assessment group with one item, reported as $7.14\%$ in data/exam_format_topic_crosswalk_2022.jsonl. This is assessment guidance, not a replacement for the syllabus.

Clean or relevant unreviewed signals from recent extracted papers include:

| Year | Question ID | Signal | |---|---|---| | 2022 | csee_041_2022_p1_q10_b | Use the quadratic formula to solve an equation obtained in a previous part. | | 2025 | csee_041_2025_p1_q10_a | Use substitution to express an equation in the form $ax^2+bx+c=0$. | | 2025 | csee_041_2025_p1_q10_b | Solve the quadratic equation obtained from the substitution step. |

These signals suggest that learners should practise both direct quadratic solving and rewriting expressions into quadratic form. The mappings are explicitly unreviewed and should not yet be treated as official past-question links.

Source And Review Notes

• Official syllabus authority: raw/syllabuses/csee/2023/csee_mathematics_syllabus_2023.pdf and the local source page CSEE Mathematics Syllabus 2023.
• Registry source: data/curriculum_map.json lists this topic as Form II, sequence 18, under the algebra and matrices hub.
• Assessment signals: data/topic_frequency_2021_2025.json, data/question_map_2021_2025.jsonl, and data/exam_format_topic_crosswalk_2022.jsonl were consulted only for unreviewed signals.
• Learner prose, worked examples, and practice tasks in this page are original draft content for review.
• Review risks: factorisation depth, graph-feature expectations, and substitution examples should be checked by a Mathematics reviewer against the official syllabus and classroom sequence before marking this page reviewed.