Quadratic Equations

Lesson

As we have seen, variables can appear in equations with no powers, ($x$`x`, $y$`y`, or $z$`z` for example) in squares and cubes, ($x^2$`x`2, $y^2$`y`2, $z^3$`z`3 for example) or even higher or rational powers ($x^5$`x`5, $y^{10}$`y`10, $z^{\frac{1}{2}}$`z`12).

The expression $3x+6$3`x`+6 has $2$2 terms, it has the $3x$3`x` and the $6$6. We already know from our work in algebra that we can collect, add, subtract, multiply and divide like terms.

A binomial expression has $2$2 terms, while a trinomial expression has $3$3 terms.

The variables, or variables, are the letters or unknowns in an expression or equation.

The coefficient** **is the number (whole, rational, irrational, real) in front of a variable.

We call the highest power of an equation the degree of the equation.

Let's look at a few examples:

$y=3x-4$`y`=3`x`−4 has degree $1$1

$y=x^2+2x+1$`y`=`x`2+2`x`+1 has degree $2$2

$y=-4x^3+3x^2-8x+16$`y`=−4`x`3+3`x`2−8`x`+16 has degree $3$3

$y=x^4-x^3-x^2+x+7$`y`=`x`4−`x`3−`x`2+`x`+7 has degree $4$4

$y=x^{10}-x^6+x^2+12$`y`=`x`10−`x`6+`x`2+12 has degree $10$10

We would define the degree of a polynomial to be equal to the **highest **power that is present in the equation.

A quadratic equation is a type of polynomial that has a degree of $2$2.

The following are all examples of quadratics:

$x^2$`x`2

$5x^2$5`x`2

$\frac{x^2}{6}+3x-7$`x`26+3`x`−7

$2x+7-18x^2$2`x`+7−18`x`2

All quadratic equations have common properties, apart from just having a degree of $2$2. Use this interactive to explore different quadratic equations. Can you see the common physical properties?

At this stage, all we really want to explore is the shape of the function.

We can interpret the solution to an equation in $2$2 different ways:

- It is the values of $x$
`x`that makes $y=0$`y`=0 (an algebraic explanation). - It is the $x$
`x`-intercepts of the graph (a graphical explanation).

Using the above applet, make the following graphs. See how many $x$`x`-intercepts each quadratic has.

- $y=x^2+6$
`y`=`x`2+6 - $y=-2\left(x+2\right)\left(x-6\right)$
`y`=−2(`x`+2)(`x`−6) - $y=\left(x+4\right)^2$
`y`=(`x`+4)2

What did you notice?

Hopefully you saw that some quadratics cross the $x$`x`-axis, some only touch the axis once and some do not cross at all. That is, they can have no, $1$1 or $2$2 $x$`x`-intercepts. We can think of these intercepts as being the solutions to the quadratic equation.

From only an equation we would have to solve it (set it equal to $0$0 and solve for $x$`x`) to find out algebraically how many solutions it has, but if we have a graph it is much easier to tell. The next few chapters will help us with the algebraic solutions.

Let's have a look at these worked examples.

Is $7x^3+3x^2$7`x`3+3`x`2 a quadratic expression?

Yes

ANo

B

Which of the following equations represent a quadratic relationship between $x$`x` and $y$`y`?

$y=1-x^2$

`y`=1−`x`2A$y=\left(x^2-1\right)^3$

`y`=(`x`2−1)3B$y=x^2+\frac{2}{x}$

`y`=`x`2+2`x`C$y=\sqrt{x^2+2}$

`y`=√`x`2+2D$y=2x^2$

`y`=2`x`2E

What relationship between $x$`x` and $y$`y` is represented by the given graph?

Linear

AQuadratic

BOther

C