Fundamentals of Business Mathematics and Statistics · Quadratic Equations
Introduction to Quadratic Equations: Definition and Standard Form
Updated 10 October 2026 · Fact-checked
A quadratic equation is an equation in one variable whose highest power is 2. Its standard form is ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0. To handle any question, move all terms to one side, identify a, b and c, then solve or frame the equation.
Understand Introduction to Quadratic Equations
An equation is a statement that two expressions are equal. The degree of an equation in one variable is the highest power of that variable. If the degree is 1, it is linear, like 2x + 3 = 7. If the degree is 2, it is a quadratic equation.
The standard form is ax² + bx + c = 0. Here x is the variable. The numbers a, b and c are called coefficients. a is the coefficient of x², b is the coefficient of x, and c is the constant term. The condition a ≠ 0 is essential. If a = 0, the x² term vanishes and the equation becomes linear.
b and c can be zero. So x² - 9 = 0 (b = 0) and 3x² + 5x = 0 (c = 0) are both quadratic. Many equations do not look quadratic at first. For example, (x + 2)(x + 3) = 20 becomes x² + 5x + 6 = 20, which is x² + 5x - 14 = 0 after you move everything to one side. Always expand and simplify before you decide.
A quadratic equation has at most two roots. A root is a value of x that makes the equation true. Business problems often lead to quadratics, such as problems on area, consecutive numbers, age, price and quantity. In a word problem, you pick a variable, write the given relation as an equation, and simplify it into standard form.
Key formulas to remember
- Standard form
- ax² + bx + c = 0, a ≠ 0
- a is the coefficient of x², b of x, c is the constant. Always keep signs with the coefficients.
- Degree test
- Highest power of x = 2 after simplification
- Expand brackets and clear fractions before checking the degree.
- Number of roots
- A quadratic equation has at most 2 roots
- The roots may be two different values, two equal values, or not real.
- Root check
- x = k is a root if a·k² + b·k + c = 0
- Substitute the value into the equation to test it.
- Consecutive integers
- x, x + 1 for integers; x, x + 2 for consecutive even or odd integers
- Useful for framing word problems on numbers.
How to solve Introduction to Quadratic Equations questions
Use this method for any question that asks you to identify, rewrite or frame a quadratic equation.
- 1Read the question and decide what is unknown. Call it x.
- 2Write the given condition as an equation using x.
- 3Expand all brackets and clear any fractions by multiplying by the common denominator.
- 4Move every term to the left side so the right side is 0.
- 5Collect like terms and arrange in the order x², x, constant.
- 6Check that the coefficient of x² is not zero. If it is, the equation is not quadratic.
- 7Read off a, b and c with their signs. For roots, substitute each option into the equation.
Quickest way: Expand, shift, check the x² term
When to use it: Use for MCQs that ask whether an equation is quadratic, or ask for the values of a, b, c or the framed equation.
- Expand brackets only as far as needed to see the x² terms.
- If x² terms cancel on both sides, the equation is not quadratic.
- Shift everything to one side and read the coefficients with signs.
- For a framed equation, compare your result with the options rather than solving fully.
- To test given roots, substitute the simplest option first and eliminate the wrong ones.
Common mistakes in Introduction to Quadratic Equations
Calling x(x + 3) = x² + 5 a quadratic equation.
The squares appear on both sides, so students assume it is quadratic without simplifying.
Fix: Expand and move all terms to one side. Here x² + 3x = x² + 5 gives 3x - 5 = 0, which is linear.
Losing the sign of b or c.
Students read coefficients before moving terms to one side.
Fix: First write the equation as ax² + bx + c = 0. For x² = 5x - 6, it becomes x² - 5x + 6 = 0, so b = -5 and c = 6.
Forgetting that a ≠ 0.
Students focus on the form and ignore the condition.
Fix: Always state a ≠ 0. In kx² + 4x + 1 = 0, k must not be 0.
Treating a missing term as a missing coefficient.
In 2x² - 8 = 0, students think b does not exist.
Fix: A missing term means its coefficient is 0. Here b = 0.
Wrong framing of consecutive numbers, such as x and x + 2 for consecutive integers.
Students mix up integers with even or odd integers.
Fix: Use x and x + 1 for consecutive integers. Use x and x + 2 only for consecutive even or consecutive odd integers.
Not checking the framed equation against the question.
Students rush and write the product or sum wrongly.
Fix: Substitute a simple trial value or reread the condition once to confirm the equation matches.
Worked examples
Example 1
Which of the following is a quadratic equation? (A) (x + 1)² = x² + 3x + 2 (B) (x + 2)(x - 1) = x² + 5 (C) (x + 2)(x - 1) = 3x + 4 (D) x³ - x(x² - 4) = 8
Show the solution
- Option A: x² + 2x + 1 = x² + 3x + 2 gives -x - 1 = 0. This is linear.
- Option B: x² + x - 2 = x² + 5 gives x - 7 = 0. This is linear.
- Option C: x² + x - 2 = 3x + 4 gives x² - 2x - 6 = 0. The highest power is 2, so it is quadratic.
- Option D: x³ - x³ + 4x = 8 gives 4x - 8 = 0. This is linear.
Answer: Option C
Example 2
The product of two consecutive positive integers is 132. Frame the quadratic equation and verify with the integers.
Show the solution
- Let the smaller integer be x. The next integer is x + 1.
- Product condition: x(x + 1) = 132.
- Expand: x² + x = 132.
- Move to one side: x² + x - 132 = 0. Here a = 1, b = 1, c = -132.
- Check: x = 11 gives 11 × 12 = 132, which is correct.
- Substituting x = 11: 121 + 11 - 132 = 0, so it satisfies the equation.
Answer: x² + x - 132 = 0; the integers are 11 and 12
Exam tips
- For 'is it quadratic' questions, always expand first. The x² terms often cancel in the trap options.
- In framing questions, check the options for the standard form and match signs of b and c carefully.
- If options give possible roots, substitute them into the equation. This is often faster than solving.
- Write consecutive numbers correctly: x, x + 1 for integers and x, x + 2 for even or odd integers.
- Do not leave any question blank. There is no negative marking, so make an informed guess after elimination.
Practice questions from Quadratic Equations
- For the quadratic equation 2x² − 7x + 6 = 0, what is the value of the discriminant and what does it say about the roots?
- For what value of k does the equation x^2 - (k + 2)x + 9 = 0 have equal roots, given that k is positive?
- A quadratic equation 2x² − 7x + 3 = 0 is solved by the quadratic formula. What is the larger root?
- For the quadratic equation 2x^2 - 5x + 3 = 0, the discriminant and the nature of its roots are:
- If one root of x^2 - 9x + k = 0 is twice the other root, what is the value of k?
Introduction to Quadratic Equations in other exams
The same ground in other exams, if you are preparing for more than one or want another angle on it.
Introduction to Quadratic Equations: frequently asked questions
What is the standard form of a quadratic equation?
The standard form is ax² + bx + c = 0, where a, b and c are real numbers and a ≠ 0. Here a is the coefficient of x², b is the coefficient of x and c is the constant term.
Why can a not be zero in a quadratic equation?
If a = 0, the x² term disappears and the equation becomes bx + c = 0, which is linear. A quadratic equation must have degree 2, so a must be non-zero.
Can b or c be zero in a quadratic equation?
Yes. Equations like x² - 16 = 0 and 2x² + 6x = 0 are quadratic. A missing term simply means its coefficient is zero.
How do I form a quadratic equation from a word problem?
Choose a variable for the unknown, write the given condition as an equation, then expand and move all terms to one side. Arrange it as ax² + bx + c = 0 and check it against the question.