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Quadratic functions

Water
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 $a×b=0$, 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 $x$, $f(x)=0$

Quadratic solution

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 $a$, $b$, $c$ ∈ ℝ. $a ≠ 0$.

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 ℝ.

Mathematics question

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.

Content © Andrew Bone. All rights reserved. Created : January 28, 2014 Last updated :January 22, 2016

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