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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?

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    $\begingroup$ A quadratic equation is simply an equation of the form $ax^2 + bx + c=0$ where $a \neq 0$. $\endgroup$
    – anak
    Jun 15, 2015 at 19:29
  • $\begingroup$ @anakhronizein So is the above equation quadratic? $\endgroup$
    – user456
    Jun 15, 2015 at 19:35
  • $\begingroup$ @PeaceSeeker It is not even an equation. $\endgroup$
    – user26486
    Jun 15, 2015 at 19:38
  • $\begingroup$ @user26486 Why is it so? $\endgroup$
    – user456
    Jun 15, 2015 at 19:38
  • $\begingroup$ @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. $\endgroup$
    – user26486
    Jun 15, 2015 at 19:40

7 Answers 7

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Yes, but the quadratic is not in x.

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

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

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  • $\begingroup$ I didn't know that it was quadratic in another variable! $\endgroup$ Jun 24, 2016 at 16:16
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if you substitute $\sqrt{x}=u$
so $$ \sqrt{x}+2x+3=u+2u^2+3$$

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  • $\begingroup$ That doesn't answer the question. Is $\sqrt{x}+2x+3$ quadratic in $x$? Is it quadratic in $u$? $\endgroup$
    – user223391
    Jun 15, 2015 at 19:32
  • $\begingroup$ @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 $u$"? $\endgroup$
    – Mico
    Jun 16, 2015 at 3:10
  • $\begingroup$ I'm saying that a good answer should include the word "quadratic". You just have an equation and no answer. $\endgroup$
    – user223391
    Jun 16, 2015 at 3:24
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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 $2(x^{1/2})^2 + x^{1/2} + 3 = 0.$

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  • $\begingroup$ But it isn't quadratic in x right? $\endgroup$
    – user456
    Jun 15, 2015 at 19:40
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    $\begingroup$ You can answer that yourself. Does it have the form $ax^2 + bx + c = 0$? $\endgroup$
    – Umberto P.
    Jun 15, 2015 at 19:44
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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 $x$:

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

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

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

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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.

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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).

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