What is a Quadratic Equation?
A quadratic equation is a polynomial equation of degree 2, written in the standard form ax² + bx + c = 0, where a ≠ 0. It appears throughout algebra, physics, and engineering — from projectile motion to profit optimization — and its solutions (roots) can be found using factoring, completing the square, or the quadratic formula.
A quadratic equation ax² + bx + c = 0 is solved using the quadratic formula x = (−b ± √(b² − 4ac)) / 2a, which gives the two values of x (real or complex) that make the equation true.
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Step-by-step worked examples
Solve x² − 5x + 6 = 0 by factoring.
Find two numbers that multiply to 6 and add to −5: −2 and −3 Factor: (x − 2)(x − 3) = 0 Set each factor to zero: x = 2 or x = 3
Solve 2x² + 3x − 2 = 0 using the quadratic formula.
Identify a = 2, b = 3, c = −2 Discriminant: b² − 4ac = 9 − (4×2×−2) = 9 + 16 = 25 x = (−3 ± √25) / (2×2) = (−3 ± 5) / 4 → x = 0.5 or x = −2
Solve x² + 4x + 4 = 0.
Identify a = 1, b = 4, c = 4 Discriminant: 16 − 16 = 0 (one repeated root) x = −4 / 2 = −2 (double root)
Flashcards
Quick quiz
Q1.What is the discriminant of x² − 5x + 6 = 0?
Q2.How many real roots does a quadratic have when the discriminant is negative?
Q3.Solve x² − 9 = 0.
Q4.In 3x² − 6x + 2 = 0, what is b?
The full card deck, worked steps and AI-tutor support for “What is a Quadratic Equation?” are in Notek — study by hand before your exam.
Common mistakes
Forgetting the ± sign, giving only one root. — Correct: The quadratic formula always yields two roots (unless the discriminant is 0).
Dividing only part of the numerator by 2a. — Correct: The entire numerator (−b ± √(b²−4ac)) must be divided by 2a.
Assuming every quadratic factors with integers. — Correct: Many quadratics require the quadratic formula because they don't factor neatly.
Ignoring a negative discriminant and reporting 'no solution'. — Correct: A negative discriminant means the roots are complex numbers, not that there's no solution at all.
FAQ
What is the quadratic formula?
x = (−b ± √(b² − 4ac)) / 2a — it solves any equation of the form ax² + bx + c = 0.
How do you calculate the roots of a quadratic equation?
Identify a, b, and c, compute the discriminant b² − 4ac, then apply the quadratic formula.
What are examples of quadratic equations?
x² − 5x + 6 = 0, 2x² + 3x − 2 = 0, and x² + 4x + 4 = 0 are all quadratic equations.
What does a negative discriminant mean?
It means the equation has no real roots — instead it has two complex conjugate roots.