One of the four parts the two axes cut the plane into, numbered I to IV, going anticlockwise from the top right.
Precisely: One of the four regions of the coordinate plane bounded by the axes: I (x > 0, y > 0), II (x < 0, y > 0), III (x < 0, y < 0) and IV (x > 0, y < 0).
How it works
Look at the signs of the two coordinates: (+, +) is I, (−, +) is II, (−, −) is III and (+, −) is IV. A point on an axis is in no quadrant.
Examples
(3, 5) is in the first quadrant, because both coordinates are positive.
(−2, 4) lies in the second quadrant: left of the y-axis and above the x-axis.
In the third quadrant both coordinates are negative, like (−1, −6).
(0, 7) is in no quadrant, because it lies on the y-axis.
Do not confuse: In the coordinate plane a quadrant is one of four regions. A quadrant of a circle is a quarter of a circle, a different meaning.
Used in
plotting points, signs of trigonometric ratios (Class 11), reflections
OpenStax: College Algebra 2e, 2.1 Key Terms (quadrant)
quadratic equation
kwod-RAT-ik i-KWAY-zhunClass 10
An equation with an x² term and no higher power of x, like 2x² − 5x + 3 = 0.
Precisely: An equation of the form ax² + bx + c = 0 with a ≠ 0 (its standard form), where a, b and c are real numbers.
How it works
Bring every term to one side to get the standard form. Then solve by factorising (splitting the middle term) or by the quadratic formula. It has at most two roots, and the discriminant tells you how many real ones.
Examples
A prayer hall whose length is one more than twice its breadth x, with area 300 m², gives the quadratic equation 2x² + x − 300 = 0.
The quadratic equation 2x² − 5x + 3 = 0 factorises as (2x − 3)(x − 1) = 0, so x = 3/2 or x = 1.
(x − 2)² + 1 = 2x − 3 is a quadratic equation: moved to one side it becomes x² − 6x + 8 = 0.
(x + 2)³ = x³ − 4 is a quadratic equation in disguise: the x³ terms cancel, leaving 6x² + 12x + 12 = 0.
Do not confuse: A quadratic polynomial (x² − 5x + 6) is an expression; a quadratic equation (x² − 5x + 6 = 0) asks when it equals 0. Its roots are the polynomial's zeroes.
Used in
areas and dimensions, speed, time and distance problems, ages and numbers puzzles, projectile paths in physics
OpenStax: Intermediate Algebra 2e, 9.1 Solve Quadratic Equations Using the Square Root Property
quadratic formula (Śrīdharāchārya's formula)
kwod-RAT-ik FOR-myoo-luhClass 10
A formula that gives the roots of any quadratic equation ax² + bx + c = 0 straight from a, b and c.
Precisely: x = (−b ± √(b² − 4ac)) ÷ 2a, the roots of ax² + bx + c = 0 (a ≠ 0) when b² − 4ac ≥ 0.
How it works
It comes from completing the square on ax² + bx + c = 0. Work out the discriminant b² − 4ac first; if it is negative there are no real roots, otherwise put its square root into the formula once with + and once with −.
Examples
For 2x² + x − 300 = 0 the quadratic formula gives x = (−1 ± √2401) ÷ 4 = (−1 ± 49) ÷ 4, so x = 12 or −12.5; the hall is 12 m by 25 m.
For x² − 2x − 1 = 0 the quadratic formula gives x = (2 ± √8) ÷ 2 = 1 ± √2.
Śrīdharāchārya (about 1025 CE) derived the quadratic formula by completing the square; Brahmagupta (598 to 665 CE) had already given a rule for ax² + bx = c.
Do not confuse: The ± means two roots, one with + and one with −. And 2a is under the whole top, not just under √(b² − 4ac).
Used in
quadratic equations that do not factorise, physics and engineering, the nature of roots