Lesson Note
INTRODUCTION
Quadratic equations are the backbone of many problems in mathematics and everyday life. They appear whenever a relationship involves a variable squared, such as in the area of a rectangle whose sides are related, in the motion of a ball thrown upward, or in the design of parabolic mirrors. Understanding how to write, manipulate, and solve these equations gives students a powerful tool for analysing real‑world situations.
In this lesson we will explore the basic form of a quadratic equation, learn how to factor quadratic expressions, and use factorisation to find the solutions of quadratic equations. We will also see how the discriminant tells us about the nature of the roots, and we will touch upon the quadratic formula as a quick alternative. By the end of the lesson students will be able to solve any quadratic equation that can be factorised, recognise the type of roots, and apply these skills to practical problems.
MAIN CONTENT
Standard Form of a Quadratic Equation
A quadratic equation is a polynomial equation of degree two. Its standard form is
where $a$, $b$, and $c$ are real numbers and $a\neq0$. The variable $x$ is the unknown we want to find. The coefficient $a$ is called the leading coefficient, $b$ is the linear coefficient, and $c$ is the constant term. The equation is set equal to zero so that we can solve for the values of $x$ that make the expression true.
Worked Example
Solve the quadratic equation $2x^{2}-4x-6=0$ by reducing to standard form and factorising.
Step 1: Identify the coefficients.
$a=2,\; b=-4,\; c=-6$.
Step 2: Factor out the common factor $2$ to simplify:
$2(x^{2}-2x-3)=0$.
Step 3: Factor the quadratic inside the parentheses:
$x^{2}-2x-3=(x-3)(x+1)$.
Step 4: Set each factor equal to zero:
$x-3=0\;\Rightarrow\;x=3$,
$x+1=0\;\Rightarrow\;x=-1$.
Answer: The solutions are $x=3$ and $x=-1$.
Practice Questions
1. Write the quadratic equation $3x^{2}+5x-2=0$ in standard form.
Answer: $3x^{2}+5x-2=0$.
2. Identify $a$, $b$, and $c$ in the equation $-x^{2}+4x+7=0$.
Answer: $a=-1,\; b=4,\; c=7$.
3. If $5x^{2}-15x+10=0$, what is the leading coefficient?
Answer: $5$.
Factorization of Quadratic Expressions
Factorisation is the process of expressing a quadratic expression as a product of two binomials. A quadratic expression of the form $ax^{2}+bx+c$ can often be written as $(mx+n)(px+q)$ where $m$, $n$, $p$, and $q$ are integers or rational numbers. The product of the first terms gives $ax^{2}$, the product of the last terms gives $c$, and the sum of the cross terms gives $bx$.
Worked Example
Factorise the quadratic expression $x^{2}+7x+12$.
Step 1: Look for two numbers that multiply to $12$ and add to $7$.
$3$ and $4$ satisfy this: $3\times4=12$ and $3+4=7$.
Step 2: Write the factors:
$(x+3)(x+4)$.
Answer: $x^{2}+7x+12=(x+3)(x+4)$.
Practice Questions
1. Factorise $x^{2}-5x+6$.
Answer: $(x-2)(x-3)$.
2. Factorise $2x^{2}+8x+6$.
Answer: $2(x+1)(x+3)$.
3. Factorise $x^{2}+x-6$.
Answer: $(x+3)(x-2)$.
Solving Quadratic Equations by Factorization
Once a quadratic equation is factorised, we can use the zero‑product property: if $(x-a)(x-b)=0$, then either $x-a=0$ or $x-b=0$. Thus the solutions are $x=a$ and $x=b$.
Worked Example
Solve $x^{2}-9=0$ by factorisation.
Step 1: Recognise a difference of squares:
$x^{2}-9=(x-3)(x+3)$.
Step 2: Set each factor to zero:
$x-3=0\;\Rightarrow\;x=3$,
$x+3=0\;\Rightarrow\;x=-3$.
Answer: $x=3$ or $x=-3$.
Practice Questions
1. Solve $x^{2}+4x+4=0$.
Answer: $x=-2$ (double root).
2. Solve $4x^{2}-12x+9=0$.
Answer: $x=\frac{3}{2}$ (double root).
3. Solve $x^{2}-6x+5=0$.
Answer: $x=1$ or $x=5$.
Discriminant and Nature of Roots
The discriminant of a quadratic equation $ax^{2}+bx+c=0$ is given by
It tells us the nature of the roots:
- If $\Delta>0$, two distinct real roots.
- If $\Delta=0$, one real root (a repeated root).
- If $\Delta<0$, two complex conjugate roots.
Worked Example
Determine the nature of the roots of $x^{2}-4x+5=0$.
Step 1: Compute the discriminant.
$\Delta=(-4)^{2}-4(1)(5)=16-20=-4$.
Step 2: Since $\Delta<0$, the equation has two complex conjugate roots.
Answer: Two complex roots.
Practice Questions
1. Find $\Delta$ for $2x^{2}+3x-5=0$.
Answer: $49$ (two real roots).
2. For $x^{2}+6x+9=0$, what is the discriminant and the type of roots?
Answer: $\Delta=0$; one repeated real root.
3. If $\Delta=25$ for a quadratic equation, how many real roots does it have?
Answer: Two distinct real roots.
Quadratic Formula (Brief)
When factorisation is difficult, the quadratic formula provides a quick way to find the roots:
This formula is derived from completing the square and works for all quadratic equations.
Worked Example
Solve $x^{2}-2x-8=0$ using the quadratic formula.
Step 1: Identify $a=1,\; b=-2,\; c=-8$.
Step 2: Compute the discriminant.
$\Delta=(-2)^{2}-4(1)(-8)=4+32=36$.
Step 3: Apply the formula.
$$x=\frac{-(-2)\pm\sqrt{36}}{2(1)}=\frac{2\pm6}{2}$$.
Step 4: Find the two solutions.
$x=\frac{8}{2}=4$ or $x=\frac{-4}{2}=-2$.
Answer: $x=4$ or $x=-2$.
Practice Questions
1. Use the quadratic formula to solve $3x^{2}+2x-1=0$.
Answer: $x=\frac{-1\pm\sqrt{4}}{3}$, i.e. $x=\frac{-1+2}{3}=\frac{1}{3}$ or $x=\frac{-1-2}{3}=-1$.
2. Solve $x^{2}+x-6=0$ using the formula.
Answer: $$x=\frac{-1\pm\sqrt{1+24}}{2}=\frac{-1\pm5}{2}$$, giving $x=2$ or $x=-3$.
3. Find the roots of $5x^{2}-20x+15=0$ with the formula.
Answer: $$x=\frac{20\pm\sqrt{400-300}}{10}=\frac{20\pm10}{10}$$, so $x=3$ or $x=1$.
SUMMARY
Quadratic equations, written as $ax^{2}+bx+c=0$, are central to many mathematical applications. By learning to factorise these expressions, students can quickly find the roots using the zero‑product property. The discriminant $b^{2}-4ac$ provides immediate insight into whether the solutions are real or complex. When factorisation is not straightforward, the quadratic formula offers a reliable alternative. Mastery of these techniques equips students with the tools to tackle a wide range of algebraic problems and to recognise the underlying patterns in real‑world scenarios.
EVALUATION
1. Solve $x^{2}+5x+6=0$ by factorisation.
2. Find the discriminant for $4x^{2}-12x+9=0$.
3. Use the quadratic formula to solve $x^{2}-4x+4=0$.
4. Determine the nature of the roots of $x^{2}+2x+5=0$.
5. Factorise $6x^{2}+15x+9$ and solve for $x$.