quadratic equation
A quadratic equation is a question of the form “for what value of x does this expression hit zero?”, where the expression is built from a squared term, a plain term, and a number. The word “quadratic” comes from the Latin for “square”, because the highest power of the unknown is two — the x is squared.
Written in symbols, a quadratic equation looks like ax^2 + bx + c = 0, where a, b, c are fixed numbers and a is not zero. The condition a ≠ 0 is essential: if a were zero, the x^2 term would vanish and you would have a mere linear equation, not a quadratic one. The numbers a, b, c are the coefficients; c on its own is the constant term.
A quadratic equation can have two real solutions, one repeated real solution, or no real solutions at all (only complex ones). This is unlike a linear equation, which always has exactly one solution. The solutions are called the roots, and you can find them by factoring, by completing the square, or by the quadratic formula.
x^2 - 5x + 6 = 0 is a quadratic equation with a = 1, b = -5, c = 6. Its roots are x = 2 and x = 3.
Check: 2^2 - 5(2) + 6 = 4 - 10 + 6 = 0. ✓
“Quadratic” means degree two, even though “quad” usually suggests four. The link is to “square” (a four-sided figure), since x^2 is the area of a square with side x.