A stream of water follows the path of a trajectory, and can be described by a quadratic equation: a parabola

A quadratic equation is a special type of polynomial, which has a variable to the order of magnitude 2. e.g. \$x^2 + 2x - 1\$ has x to the order of magnitudes 2 and 1.

Zero factor property: if \$aĆb=0\$, then it follows that either \$a=0\$ and/or \$b=0\$.

The root or zero of a function is the value for x which causes the function to equal zero.

If a quadratic can be reduced to two factors equal to zero, then the zeros of the function can be identified.

e.g. \$(x+2)(x-3)=0\$ has two solutions: \$x=-2\$ and \$x=3\$. For these values of \$x\$, \$f(x)=0\$

The solution to the quadratic equation (binomial solution) \$ax^2 + bx + c = 0\$ is:

\$\$x = {-b ± √{b^2 - 4ac}}/{2a}\$\$

For a quadratic in the form \$y = ax^2 + bx + c\$, the axis of symmetry is \$x = {-b}/{2a}\$, and the vertex is located at: \$({-b}/{2a}, f{-b}/{2a})\$.

\$({-b}/{2a}, f{-b}/{2a})\$

### Special cases

For the quadratic equation \$ax^2 + bx + c = 0\$:

(i) If b = 0, \$ax^2 + c = 0\$, so \$x = ±√{{-c}/a}\$

(ii) If c = 0, \$ax^2 + bx = 0\$, so \$x(ax + b) = 0\$, so \$x = 0\$, or  \$x = {-b}/a\$

(iii) If b and c = 0, \$x^2 = 0\$, so \$x = 0\$

## Completing the Square

A square of factors expands out to: \$(x ± p)^2 = (x ± p)(x ± p) = x^2 ± 2px + p^2\$

A general quadratic, \$ax^2 + bx + c = 0\$ can be rewritten as: \$x^2 + b/ax + c/a = 0\$

\$x^2 + b/ax = -c/a\$

\$x^2 + 2x⋅b/{2a} = -c/a\$

\$x^2 + 2x⋅b/{2a} + {(b/{2a})}^2 = {(b/{2a})}^2 - c/a\$

The part on the left is of the form \$ x^2 ± 2px + p^2 = (x ± p)^2\$, where \$p = {(b/{2a})}\$

∴ \$(x + b/{2a})^2 = {(b/{2a})}^2 - c/a\$

\$x + b/{2a} = ±√{{b^2 - 4ac}/{4a^2}}\$

\$x = -b/{2a} ±√{{b^2 - 4ac}/{4a^2}}\$

\$x = {-b ± √{b^2 - 4ac}}/{2a}\$

### Factorization

The factors may be found by using the quadratic formula, which supplies answers \$p\$ and \$q\$ for equations with two roots. The factors are: \$(x+q)(x+q)\$.

### Viète's Theorem

\$x_1 + x_2 = -b/a\$ and \$x_1⋅x_2 = c/a\$

where \$x_1\$ and \$x_2\$ are the two solutions to \$ax^2+b+c=0\$, where \$a\$, \$b\$, \$c\$ ā ā. \$a ≠ 0\$.

### Discriminant

For a quadratic equation, \$ax^2 + bx +c\$, the discriminant is: \$Δ = b^2 - 4ac\$.

The number of roots (solutions) a quadratic has depends on the sign of the discriminant.

The discriminant is the part of the quadratic solution which appears under the root sign. If the discriminant is negative, the root has no solution in ā.

The discriminant determines how many solutions for \$x\$ there are when the function equals zero (i.e. how many x-axis intercepts there are).

\$Δ > 0\$ : 2 solutions for x when the function equals zero. The parabola crosses the x-axis two times.

\$Δ = 0\$ : 1 solution for x. The parabola touches the x-axis without crossing it.

\$Δ > 0\$ : 0 solutions for x. The parabola does not touch or cross the x-axis.

## Site Index

### Latest Item on Science Library:

The most recent article is:

Air Resistance and Terminal Velocity

View this item in the topic:

Mechanics

and many more articles in the subject:

### Environment

Environmental Science is the most important of all sciences. As the world enters a phase of climate change, unprecedented biodiversity loss, pollution and human population growth, the management of our environment is vital for our futures. Learn about Environmental Science on ScienceLibrary.info.

### Great Scientists

#### Carl Linnaeus

1707 - 1778

Carl Linnaeus was a prolific writer, publishing books, lavishly illustrated, throughout his life. Through his travels, studies and collections, he developed a system of taxonomic nomenclature which is the basis of the modern system.

### Quote of the day...

"I knew Descartes," said Isaac Barrow. "René', I used to say. 'Have you got des cartes?' Then after a few tankards, he would say: 'Don't try to cheat me, Wheelie-boy. Cogito ergo sum...' - that's classical pidgin for 'I can think so I can add up as well as the next man'."