In mathematics, a polynomial equation of second degree (that is, an equation containing as its highest power the square of a variable, such as x2). The general formula of such equations is
ax2 + bx + c = 0
in which a, b, and c are real numbers, and only the coefficienta cannot equal 0.
In coordinate geometry, a quadratic function represents a parabola.
Some quadratic equations can be solved by factorization (see factor (algebra)), or the values of x can be found by using the formula for the general solution
x = [−b + √(b2 − 4ac)]/2a or
x = [−b − √(b2 − 4ac)]/2a
Depending on the value of the discriminant, b2 − 4ac, a quadratic equation has two real, two equal, or two complex roots (solutions). When b2 − 4ac > 0, there are two distinct real roots. When b2 − 4ac = 0, there are two equal real roots. When b2 − 4ac < 0, there are two distinct complex roots.
Solving quadratics by factorizing If a quadratic equation can be factorized, it can be solved. For example, to solve x2 + x − 6 = 0, the equation should be factorized to give:
(x + 3)(x − 2) = 0
There are now two brackets multiplied together to give 0. This means that one of them must be 0. So either x + 3 = 0 or x − 2 = 0. Therefore, x = −3 or x = 2.
Drawing the graphs of quadratics Quadratic equations always plot parabolas, whereas linear equations plot straight lines. For example, to draw the graph of y = x2 − 2x − 4, the table should be completed:
and then the graph plotted:
From the graph, the solution to the equation y = x2 − 2x − 4 is
x = −1.2 or x = 3.2
Completing the square can solve other quadratics.
Multiplying out brackets and factorizing
Out for the Count – Problem Solving Using Quadratic Equations
Algebra: simultaneous equations
Factorizing quadratic equations
Simplification of surds
Quadratic, Cubic, and Quartic Equations
SOS Mathematics: Algebra
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