Quadratic Equations • Topic 1 of 3

Standard Form of a Quadratic Equation

What is the standard form of a quadratic equation? A quadratic equation in the variable \(x\) is any equation that can be written in the form \(ax^{2} + bx + c = 0\), where \(a\), \(b\) and \(c\) are real numbers and, crucially, \(a \neq 0\). The highest power of the variable is exactly 2 — the word "quadratic" comes from the Latin quadratus, meaning "square".

The three coefficients. In \(ax^{2}+bx+c=0\): \(a\) is the leading coefficient (coefficient of \(x^{2}\)), \(b\) is the coefficient of the linear term \(x\), and \(c\) is the constant term. Note that \(b\) or \(c\) may be zero — only \(a\) is forbidden from being zero. So \(x^{2}-9=0\) (here \(b=0\)) and \(2x^{2}-4x=0\) (here \(c=0\)) are both perfectly good quadratic equations.

Why must \(a \neq 0\)? If \(a=0\) the \(x^{2}\) term vanishes and the equation collapses to \(bx+c=0\), which is linear (degree 1), not quadratic. The condition \(a\neq0\) is what guarantees a genuine degree-2 equation with at most two roots.

How to check whether an equation is quadratic. Bring every term to one side so the other side is 0, expand all brackets, simplify, and then look at the highest power of \(x\) that actually survives. If after simplification the highest surviving power is 2 (with a non-zero coefficient), the equation is quadratic — even if it did not look like it at first. For example, \((x-2)^{2}=5\) and \(x(x+1)=3\) both reduce to standard quadratics, while \((x+1)^{3}=x^{3}\) surprisingly reduces to a quadratic because the \(x^{3}\) terms cancel.

Converting to standard form — the routine:

  • Expand every bracket and clear fractions (multiply through by the common denominator).
  • Transfer all terms to the left side so the right side becomes 0.
  • Combine like terms and arrange in descending powers of \(x\): \(x^{2}\) term first, then \(x\), then constant.
  • Read off \(a\), \(b\), \(c\).

Equations that are NOT quadratic:

  • \(x^{3}+2x+1=0\) — degree 3, a cubic.
  • \(2x+5=0\) — degree 1, linear.
  • \(x+\dfrac{1}{x}=4\) looks non-polynomial, but multiplying by \(x\) gives \(x^{2}-4x+1=0\), which is quadratic (with \(x\neq0\)).

Mathematical formulation of word problems. A large part of this chapter is translating a real situation into \(ax^{2}+bx+c=0\). The standard technique: (1) name the unknown with a variable, say \(x\); (2) express every other quantity in the problem in terms of \(x\); (3) use the given relationship (a product, an area, a sum of squares, a time difference) to write one equation; (4) simplify to standard form. Two recurring patterns are worth memorising:

  • Consecutive integers: represent them as \(x\) and \(x+1\) (or \(x-1, x, x+1\) for three). "Product is 56" becomes \(x(x+1)=56\Rightarrow x^{2}+x-56=0\).
  • Rectangle / area: if the length is \(p\) more than the breadth, take breadth \(=x\), length \(=x+p\); area gives \(x(x+p)=\text{Area}\).

Common mistakes to avoid: forgetting to move everything to one side before reading \(a,b,c\); dropping the negative sign on \(b\) or \(c\) (in \(3x^{2}-2x+7=0\), \(b=-2\), not \(2\)); and assuming any equation with an \(x^{2}\) is quadratic without checking that the \(x^{2}\) term truly survives after simplification.

Real-life analogy. Throw a ball upward and its height after \(t\) seconds follows \(h=ut-\tfrac12 gt^{2}\) — a quadratic in \(t\). Asking "when does it hit the ground?" means setting \(h=0\), i.e. solving a quadratic. The same equations model the cross-section of a satellite dish, the cable of a suspension bridge, and profit-versus-price in business.

Anatomy of the standard form a x squared plus b x plus c equals 0 with each coefficient labelled Standard form of a quadratic equation a x^2 + b x + c = 0 a: x^2 term a ≠ 0 b: x term may be 0 c: constant may be 0 If a = 0 the x^2 term vanishes and it becomes linear (b x + c = 0). Converting a bracketed equation to standard form in three steps Converting to standard form 2x^2 + 5 = 3x bring all terms to the left 2x^2 − 3x + 5 = 0 read off the coefficients descending powers of x a = 2, b = −3, c = 5 A thrown ball traces a parabola, the shape modelled by a quadratic equation Why "quadratic"? A parabolic path time height highest point launch lands (h = 0) Height h = ut − ½gt^2 is a quadratic in t; setting h = 0 solves a quadratic.
Explore it yourself
Solved Examples
1
Worked Example
Identify \(a\), \(b\), and \(c\) for the quadratic equation \(3x^{2} - 2x + 7 = 0\).
Solution
  1. Step 1: Compare with the standard form \(ax^{2} + bx + c = 0\).
  2. Step 2: Coefficient of \(x^{2}\) is 3, so \(a = 3\).
  3. Step 3: Coefficient of \(x\) is \(-2\), so \(b = -2\) (keep the sign).
  4. Step 4: Constant term is 7, so \(c = 7\).

Answer: \(a = 3,\ b = -2,\ c = 7\)

2
Worked Example
Write \((x + 2)^{2} = 9\) in standard quadratic form and identify \(a, b, c\).
Solution
  1. Step 1: Expand the left side: \((x + 2)^{2} = x^{2} + 4x + 4\).
  2. Step 2: The equation becomes \(x^{2} + 4x + 4 = 9\).
  3. Step 3: Move every term to the left: \(x^{2} + 4x + 4 - 9 = 0\).
  4. Step 4: Simplify: \(x^{2} + 4x - 5 = 0\).
  5. Step 5: Compare: \(a = 1,\ b = 4,\ c = -5\).

Answer: \(x^{2} + 4x - 5 = 0\); \(a=1,\ b=4,\ c=-5\)

3
Worked Example
Check whether \(x(x + 1) = 5\) is a quadratic equation. If so, write it in standard form.
Solution
  1. Step 1: Expand the left side: \(x^{2} + x = 5\).
  2. Step 2: Move the constant to the left: \(x^{2} + x - 5 = 0\).
  3. Step 3: The highest power of \(x\) is 2 and its coefficient \(a = 1 \neq 0\).
  4. Step 4: Therefore the equation is quadratic, with \(a=1,\ b=1,\ c=-5\).

Answer: Yes; standard form \(x^{2} + x - 5 = 0\)

4
Worked Example
Check whether \((x - 2)^{2} + 1 = 2x - 3\) is a quadratic equation.
Solution
  1. Step 1: Expand the left side: \(x^{2} - 4x + 4 + 1 = x^{2} - 4x + 5\).
  2. Step 2: The equation becomes \(x^{2} - 4x + 5 = 2x - 3\).
  3. Step 3: Bring all terms to the left: \(x^{2} - 4x + 5 - 2x + 3 = 0\).
  4. Step 4: Combine like terms: \(x^{2} - 6x + 8 = 0\).
  5. Step 5: Highest power is 2 with \(a = 1 \neq 0\), so it is quadratic.

Answer: Yes; standard form \(x^{2} - 6x + 8 = 0\)

5
Worked Example
Determine whether \((x + 1)^{3} = x^{3} - 1\) is a quadratic equation.
Solution
  1. Step 1: Expand the left side: \((x+1)^{3} = x^{3} + 3x^{2} + 3x + 1\).
  2. Step 2: The equation is \(x^{3} + 3x^{2} + 3x + 1 = x^{3} - 1\).
  3. Step 3: Subtract \(x^{3}\) from both sides — the cubic terms cancel: \(3x^{2} + 3x + 1 = -1\).
  4. Step 4: Bring all terms left: \(3x^{2} + 3x + 2 = 0\).
  5. Step 5: Highest surviving power is 2 with \(a = 3 \neq 0\), so it IS quadratic.

Answer: Yes; it reduces to \(3x^{2} + 3x + 2 = 0\)

6
Worked Example
Is \(x + \dfrac{1}{x} = 3\) (where \(x \neq 0\)) a quadratic equation? Write its standard form.
Solution
  1. Step 1: Multiply every term by \(x\) to clear the fraction: \(x \cdot x + x \cdot \dfrac{1}{x} = 3x\).
  2. Step 2: This gives \(x^{2} + 1 = 3x\).
  3. Step 3: Move all terms to the left: \(x^{2} - 3x + 1 = 0\).
  4. Step 4: Highest power 2, \(a = 1 \neq 0\), so it is quadratic.

Answer: Yes; standard form \(x^{2} - 3x + 1 = 0\)

7
Worked Example
For what value of \(k\) is the equation \((k - 3)x^{2} + 5x - 2 = 0\) not a quadratic equation?
Solution
  1. Step 1: An equation is quadratic only when the coefficient of \(x^{2}\) is non-zero.
  2. Step 2: Here the coefficient of \(x^{2}\) is \(k - 3\).
  3. Step 3: It fails to be quadratic when \(k - 3 = 0\).
  4. Step 4: Solve: \(k = 3\). (For any other value of \(k\) it remains quadratic.)

Answer: \(k = 3\)

8
Worked Example
The product of two consecutive positive integers is 156. Form a quadratic equation for the smaller integer \(x\).
Solution
  1. Step 1: Let the smaller integer be \(x\); the next consecutive integer is \(x + 1\).
  2. Step 2: Their product is 156: \(x(x + 1) = 156\).
  3. Step 3: Expand: \(x^{2} + x = 156\).
  4. Step 4: Move all terms left: \(x^{2} + x - 156 = 0\).

Answer: \(x^{2} + x - 156 = 0\)

9
Worked Example
The length of a rectangular plot is 3 metres more than its breadth, and its area is 154 m². Form a quadratic equation taking the breadth as \(x\).
Solution
  1. Step 1: Let the breadth be \(x\) metres; then the length is \(x + 3\) metres.
  2. Step 2: Area = length × breadth, so \(x(x + 3) = 154\).
  3. Step 3: Expand: \(x^{2} + 3x = 154\).
  4. Step 4: Write in standard form: \(x^{2} + 3x - 154 = 0\).

Answer: \(x^{2} + 3x - 154 = 0\)

10
Worked Example
The sum of the squares of two consecutive natural numbers is 313. Form the quadratic equation, taking the smaller number as \(x\).
Solution
  1. Step 1: Let the smaller number be \(x\); the next is \(x + 1\).
  2. Step 2: Sum of their squares: \(x^{2} + (x + 1)^{2} = 313\).
  3. Step 3: Expand: \(x^{2} + x^{2} + 2x + 1 = 313\).
  4. Step 4: Combine: \(2x^{2} + 2x + 1 = 313\).
  5. Step 5: Move 313 left: \(2x^{2} + 2x - 312 = 0\).
  6. Step 6: Divide throughout by 2: \(x^{2} + x - 156 = 0\).

Answer: \(x^{2} + x - 156 = 0\)

11
Worked Example
A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, it would have taken 3 hours more. Taking the speed as \(x\) km/h, form the quadratic equation. (Use time = distance ÷ speed.)
Solution
  1. Step 1: Let the speed be \(x\) km/h. Time at this speed \(= \dfrac{480}{x}\) hours.
  2. Step 2: At a reduced speed \(x - 8\), the time is \(\dfrac{480}{x - 8}\) hours, which is 3 hours more.
  3. Step 3: Set up the relation: \(\dfrac{480}{x - 8} - \dfrac{480}{x} = 3\).
  4. Step 4: Multiply through by \(x(x - 8)\): \(480x - 480(x - 8) = 3x(x - 8)\).
  5. Step 5: Simplify the left: \(480x - 480x + 3840 = 3x^{2} - 24x\), i.e. \(3840 = 3x^{2} - 24x\).
  6. Step 6: Divide by 3 and rearrange: \(x^{2} - 8x - 1280 = 0\).

Answer: \(x^{2} - 8x - 1280 = 0\)

12
Worked Example
Rohan's mother is 26 years older than him. The product of their ages (in years) 3 years from now will be 360. Taking Rohan's present age as \(x\), form the quadratic equation.
Solution
  1. Step 1: Let Rohan's present age be \(x\) years; his mother's present age is \(x + 26\) years.
  2. Step 2: After 3 years: Rohan \(= x + 3\), mother \(= x + 29\).
  3. Step 3: Product of their future ages is 360: \((x + 3)(x + 29) = 360\).
  4. Step 4: Expand: \(x^{2} + 29x + 3x + 87 = 360\), i.e. \(x^{2} + 32x + 87 = 360\).
  5. Step 5: Move 360 left: \(x^{2} + 32x - 273 = 0\).

Answer: \(x^{2} + 32x - 273 = 0\)

Key Points

  • Standard form: \(ax^{2} + bx + c = 0\), where \(a \neq 0\)
  • \(a, b, c\) are real numbers; \(a\) multiplies \(x^{2}\), \(b\) multiplies \(x\), \(c\) is the constant
  • Either \(b\) or \(c\) may be 0, but \(a\) must never be 0 (else it becomes linear)
  • To check if an equation is quadratic, expand and simplify; it is quadratic only if the highest surviving power of \(x\) is 2
  • To form a word-problem equation: name the unknown \(x\), express other quantities in terms of \(x\), use the given relation, then reduce to standard form
  • Keep the signs of \(b\) and \(c\) when reading coefficients (e.g. in \(3x^{2}-2x+7=0\), \(b=-2\))
  • Degree of a quadratic equation is always 2
Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.The standard form of a quadratic equation is:
Explanation: General quadratic form.
Q2.The degree of a quadratic equation is:
Explanation: Highest power $2$.
Q3.The roots of $(x-1)(x+2)=0$ are:
Explanation: Set each factor to $0$.
Q4.In $ax^2+bx+c=0$, we require:
Explanation: Otherwise it is not quadratic.
Practice more MCQs → 📝 Assignment · 35 marks