Quadratic equation

From RationalWiki

NOTE: This article or section is (sadly) a serious treatment of the subject matter. Unlike most of RationalWiki, it lacks sarcasm, satire, or humor in general.

x + 3 = 7 x = ?

This article/section deals with mathematical concepts appropriate for a student in early high school.

A quadratic equation is an equation that asserts that some 2^nd degree ("quadratic") polynomial in one variable is equal to zero. Solving such an equation means finding the values of the independent variable that cause the polynomial value to be zero. Those values are the "solutions" or "roots" of the equation.

By the fundamental theorem of algebra, any n^th degree polynomial can be factored into n 1^st degree polynomials, and hence an n^th degree equation has n roots, if multiplicities are counted properly. In the case of quadratic equations (n=2), this means that a quadratic polynomial

ax² + bx + c = 0

can be factored into a form

a(x − r₁)(x − r₂) = 0

from which the roots can be seen to be r₁ and r₂.

[edit] Solving a quadratic equation

There is a completely deterministic (exact, with no guesswork) way to find the roots of a quadratic equation. (In fact, this can also be done for cubic (n=3) and quartic (n=4) equations, but no higher. These methods are rather complicated.) The usual way to do this is with the traditional technique known as "completing the square", but that is a pain in the ass for equations more complex than "completing the square" homework examples. A more direct way is this:

We know that

a(x − r₁)(x − r₂) = a(x² − [r₁ + r₂]x + r₁r₂)

If that is known to be equal to

ax² + bx + c

then we must have

r₁ + r₂ = − b / a

r₁r₂ = c / a

Since we know the sum of the roots, their average can be calculated just by dividing it by two. If we know the average of two numbers, we know that one of them is higher than the average by some amount, and the other is lower than the average by the same amount.

So, the average is − b / 2a, and

r₁ = − b / 2a + D

r₂ = − b / 2a − D

for some D. Multiplying, we get

But this is c/a, so

or

or

Substituting this value for D into our two formulas for r₁ and r₂, we get:

Since the two fractions have the same denominator, they can easily be added, and because the two formulas differ only by a + and - sign in one place, they can be combined into one representation, from which we get the celebrated quadratic formula:

The two signs of the square root provide the two roots of the equation.

[edit] The discriminant

The discriminant is a big word for the gobbledygook under the square root, i.e. b² − 4ac. If that thing is positive, the quadratic equation has two distinct real solutions. If it's zero, it has exactly one real solution (due to the amazing property of zero being its own negative). If its negative, they teach you in kindergarden (or shortly afterwards) that there is no solution. That's actually true as long as you are looking only for real solutions. However, there are two distinct complex solutions, which in this case is not a synonym for "complicated". They are non-real, i.e. their imaginary part is non-zero. One may argue if non-real is the same as unreal. Long story short, if you need to know if there are any real solutions to your quadratic equation, it's sufficient to compute the discriminant and check its sign.

[edit] Parabolas

Note that if the quadratic polynomial ax² + bx + c is considered as a function of x, the solutions to the quadratic equation are the points where the graph of the function, given by the equation y = ax² + bx + c, crosses the x axis. These graphs make up the entire family of parabolas, themselves a subset of the conic sections.

If the discriminant is zero, then the corresponding parabola touches the x-axis in one point. If it is negative, the parabola doesn't intersect the x-axis at all.

Mathematics Articles on RationalWiki
	-  40gon (Fun)  -  Complex numbers  -  Conservapedian mathematics  -  Delta function  -  Fermat's last theorem  -  Fibonacci sequence  -  Golden Ratio  -  Groups  -  Gödel's incompleteness theorems  -  Information  -  Irrational number  -  Mathematics  -  Metric system  -  ln  -  Pentagon  -  Phli (fun)  -  Probability (Conservapedia)  -  Pyramid  -  Recursion  -  Rene Descartes  -  Statistics  -  TeX  -  Zero  -