quadratic equation form maximum solutions

Question

My Pearson intermediate algebra book has a "concept check" question in its section on solving equations by using quadratic methods. These questions are supposed to highlight fundamental concepts that indicate full or poor understanding of the subject. The question asks:

a. True or false? The maximum number of solutions that a quadratic equation can have is 2. b. True or false? The maximum number of solutions that an equation in quadratic form can have is 2.

The answers are listed as a. true and b. false.

I'm having difficulty searching for information on this point because search results yield explanations of how to determine the number of solutions based on the discriminant, but don't seem to get into why an equation in quadratic form is not necessarily a quadratic equation, or why it wouldn't have the same of maximum number of solutions as a quadratic equation. I'm also not finding an explanation anywhere in the text, which is mostly examples and the phrase "equation in quadratic form" is nowhere to be found.

Answer 1

I googled "quadratic in form" and found the following explanation from Paul's online notes:http://tutorial.math.lamar.edu/Classes/Alg/ReducibleToQuadratic.aspx

For example, an equation like $x^4+12x^2-74=0$ is an equation in quadratic form.

Now, you are asked the following questions:

1) How many solutions can a quadratic equation have? The answer is not more than two. To see this, write down the formula for roots of the quadratic equation $ax^2+bx+c=0$, which is:$$x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$$

Therefore there are at most two roots, given by $\dfrac{-b + \sqrt{b^2-4ac}}{2a}$ and $\dfrac{-b - \sqrt{b^2-4ac}}{2a}$. Of course, if the discriminant $b^2-4ac$ is zero, there is only one solution.

1) How many solutions can an equation in quadratic form have? The answer is can be more than two. To see this, look at the equation $x^4-3x^2+2=0$. This is an equation in quadratic form, which we write as $X^2-3X+2=0$, where $X=x^2$,and get $X = 1,2$ and the solutions are given by $1,-1,\sqrt{2},-\sqrt{2}$, which is more than two. I think this answers your question.

Answer 2

A quadratic equation is a special type of an equation in quadratic form.

Generally speaking, an equation in quadratic form involves a $\text{thing}$ and that $\text{thing squared}$. Slightly more mathematically, that's:

$$a (\text{thing})^2 + b (\text{thing}) + c = 0$$

In general, "$\text{thing}$" can represent basically anything we want. But a quadratic equation has that form with the additional restriction that $\text{thing} = x$. Or instead of $x$ it can be whatever the variable is... $y$, $z$, $t$, whatever.

A quadratic equation can have at most two solutions (and actually always has exactly two complex solutions - a fact you may learn later in your studies).

But an equation in quadratic form is a more general object and we can't necessarily say that it has at most two solutions.

An example of an equation in quadratic form would be $a(\text{thing})^2 + b(\text{thing}) + c = 0$, where $\text{thing} = x^2$ instead of just $x$. Then we have $(\text{thing})^2 = (x^2)^2 = x^4$, the general equation becomes

$$a x^4 + bx^2 + c = 0,$$

and this equation has at most four solutions (and actually, exactly four complex solutions as you may learn later).

Another and more complicated example would be if $\text{thing} = x^{1/3}.$ Then $(\text{thing})^2 = (x^{1/3})^2 = x^{2/3},$ and the equation becomes:

$$ax^{2/3} + bx^{1/3} + c = 0$$

Answer 3

$$ax^n +bx^m + c = 0$$ As far has the ratio $\frac{n}{m}$ is $2$, this is a quadratic form

The maximum number of solutions a quadratic has is $2$, though it can have a single root if $a ≠ 0$ and discriminate $\Delta = 0$

The number of solutions a quadratic form has depends on its degree $n$ , if $n \in \mathbb{Z}$ and $n \gt m$ then it has $n$ roots or $2p $ roots if $n \in \mathbb{Q}$ , $p \cdot n \in \mathbb{Z}$ and $p\cdot n \gt p\cdot m$, $pm =1$