Is $x^{\frac{1}{2}}+ 2x+3=0$ a quadratic equation

Is $$x^{\frac{1}{2}}+ 2x+3=0$$ considered a quadratic equation?

Should the equation be in the form $$ax^2+bx+c=0$$ to be considered quadratic?

• A quadratic equation is simply an equation of the form $ax^2 + bx + c=0$ where $a \neq 0$.
– anak
Jun 15, 2015 at 19:29
• @anakhronizein So is the above equation quadratic? Jun 15, 2015 at 19:35
• @PeaceSeeker It is not even an equation. Jun 15, 2015 at 19:38
• @user26486 Why is it so? Jun 15, 2015 at 19:38
• @PeaceSeeker See the definition on Wikipedia (here). An equation is an equality containing one or more variables. Your expression does not have a $=$ sign and it does not represent an equality. Jun 15, 2015 at 19:40

Yes, but the quadratic is not in x.

$2 (\sqrt x )^2 + (\sqrt x ) + 3 = 0$ can be considered to be quadratic equation in $(\sqrt x ).$

• I (not OP) edited it. Is it ok now? Jun 15, 2015 at 21:08
• Thanks,sorry not replied before. Sep 19, 2015 at 2:06

That would not be considered quadratic in $x$, but you can let $u=\sqrt{x}$ to get $2u^2+u+3$. It WOULD be quadratic in $u$.

• I didn't know that it was quadratic in another variable! Jun 24, 2016 at 16:16

if you substitute $\sqrt{x}=u$
so $$\sqrt{x}+2x+3=u+2u^2+3$$

• That doesn't answer the question. Is $\sqrt{x}+2x+3$ quadratic in $x$? Is it quadratic in $u$?
– user223391
Jun 15, 2015 at 19:32
• @avid19 - What do you mean by "is it quadratic in $u$"? Are you having difficulties answering the question, "Is $2u^2+u+3=0$ quadratic in $u$"?
– Mico
Jun 16, 2015 at 3:10
• I'm saying that a good answer should include the word "quadratic". You just have an equation and no answer.
– user223391
Jun 16, 2015 at 3:24

Equations need equal signs, just like sentences need verbs. Just as the equation $ax^2 + bx + c = 0$ is quadratic in $x$, the equation $x^{1/2} + 2x + 3 = 0$ is quadratic in $x^{1/2}$ since it can be written in the form $2(x^{1/2})^2 + x^{1/2} + 3 = 0.$

• But it isn't quadratic in x right? Jun 15, 2015 at 19:40
• You can answer that yourself. Does it have the form $ax^2 + bx + c = 0$? Jun 15, 2015 at 19:44

I think the phrase "quadratic in $x$" is short for "a quadratic polynomial in $x$". This is not a polynomial, so it is not a quadratic polynomial.

However, note that from this equation we can derive another equation that is quadratic in $x$:

$$x^{1/2} = - (2x + 3)$$

$$x = (2x+3)^2$$

$$4x^2 + 11 x + 9 =0$$

In order for an equation to be quadratic, it needs to be in either the standard form $$ax^2 + bx + c = 0$$, the vertex form (which is easy when dealing with its parabola): $$a(x - h)^2 + k = 0$$ where $h$ is the x-coordinate of the vertex and $k$ is the y-coordinate of the vertex; the factored form $$a(x - h)(x - k) = 0$$, or the intercept form $$a(x - p)(x - q) = 0$$ where $p$ is the x-coordinate of an intercept and $q$ is the y-coordinate of the intercept. Your first equation isn't quadratic, since the degree is $\frac 12$, which is represented as $\sqrt x$ (as some people said already). The equation has to have a degree of 2 in order for it to be quadratic.

The equation $f(x)=2x+\sqrt x+3$ is not a quadratic equation since there is no power of $x$ that is $2$ and not all of the powers of $x$ are integers (the square root term).

If we make the transformation $u=\sqrt x$, we get $f(u)=2u^2+u+3$ which is a quadratic equation since the powers are integers and it is of degree $2$.

However, there is an issue in doing this substitution in that extraneous solutions are found (due to plus/minus of square roots).