Coaching Centre
Quadratic identities
The difference between an identity and an equation, equating coefficients, and finding the quadratic that passes through three given points.
Identity or equation?
An equation is true only for particular values: $x^2 = 4$ is true only when $x = \pm 2$. An is true for every value of $x$, such as $(x + 1)^2 = x^2 + 2x + 1$. We sometimes write $\equiv$ for an identity.
Find $a$, $b$ and $c$ if $3x^2 - 5x + 1 \equiv a(x - 1)^2 + b(x - 1) + c$.
The quadratic through three points
A quadratic has three unknown coefficients, so three points determine it exactly (as long as no two share an $x$-value and the three are not on a line).
Drag A, B and C. There is always exactly one parabola through them. Line them up and see what happens to a.
- Put all three points on a straight line. What is $a$, and why?
- Find three points that give $a = 1$ and $c = 0$.
Find the quadratic $y = ax^2 + bx + c$ through $(0, 3)$, $(1, 2)$ and $(2, 5)$.
Which is an identity?
The difference of squares is true for every $x$.
If $x^2 + 6x + k \equiv (x + 3)^2 - 5$, find $k$.
$(x + 3)^2 - 5 = x^2 + 6x + 4$, so $k = 4$.
$(x + 2)^2 + k \equiv x^2 + 4x + 9$. Find $k$.
$(x + 2)^2 = x^2 + 4x + 4$, so $k = 9 - 4 = 5$.
$a(x + 1)^2 + b \equiv 3x^2 + 6x + 10$. Find $b$.
Match $x^2$: $a = 3$. Then $3(x^2 + 2x + 1) + b$ has constant $3 + b = 10$.
How many points are needed to fix a quadratic?
There are three unknown coefficients $a$, $b$ and $c$.
The quadratic $y = ax^2 + bx + c$ passes through $(0, 1)$, $(1, 4)$ and $(-1, 0)$. Find $a$.
$c = 1$, $a + b = 3$ and $a - b = -1$, so $a = 1$ (and $b = 2$).
Find $p$ if $x^2 + px + 9 \equiv (x - 3)^2$.
$(x - 3)^2 = x^2 - 6x + 9$.
Which is an equation rather than an identity?
$3x = 12$ is true only when $x = 4$.