What is the square root of ax^2+bx+c?

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What is the square root of ax^2+bx+c?

$\sqrt{a x^{2} + b x + c} = ?$ $sqrt(ax^2+bx+c)=?$ i tried myself and got $a^{2} x + \sqrt{c}, a^{2} x + \frac{b}{2}$ $a^(2)x+sqrt(c),a^(2)x+b/2$ But i do not think That is exactly right.

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Answer 1 · 2018-02-02T05:38:54

$\sqrt{a x^{2} + b x + c} = \sqrt{a} x + \sqrt{c}$ $sqrt(ax^2+bx+c)=sqrt a" "x +sqrt c$ , as long as $a$ $a$ and $c$ $c$ aren't negative, and $b = \pm 2 \sqrt{a c} .$ $b=+-2sqrt(ac).$

Explanation:

If $a x^{2} + b x + c$ $ax^2+bx+c$ is a perfect square, then its square root is $p x + q$ $px+q$ for some $p$ $p$ and $q$ $q$ (in terms of $a, b, c$ $a, b, c$ ).

$a x^{2} + b x + c = {(p x + q)}^{2}$ $ax^2+bx+c = (px+q)^2$
$a x^{2} + b x + c = p^{2} x^{2} + 2 p q x + q^{2}$ $color(white)(ax^2+bx+c) =p^2" "x^2 + 2pq" "x + q^2$

So, if we are given $a$ $a$ , $b$ $b$ , and $c$ $c$ , we need $p$ $p$ and $q$ $q$ so that

$p^{2} = a$ $p^2=a$ ,
$2 p q = b$ $2pq = b$ , and
$q^{2} = c$ $q^2=c$ .

Thus,

$p = \pm \sqrt{a}$ $p=+-sqrt a$ ,
$q = \pm \sqrt{c}$ $q=+-sqrt c$ , and
$2 p q = b$ $2pq = b$ .

But wait, since $p = \pm \sqrt{a}$ $p= +-sqrta$ and $q = \pm \sqrt{c}$ $q=+-sqrtc$ , it must be that $2 p q$ $2pq$ is equal to $\pm 2 \sqrt{a c}$ $+-2sqrt(ac)$ as well, so $a x^{2} + b x + c$ $ax^2+bx+c$ will only be a perfect square when $b = \pm 2 \sqrt{a c} .$ $b=+-2sqrt(ac).$ (Also, in order to have a square root, $a$ $a$ and $c$ $c$ must both be $\geq 0$ $ge 0$ .)

So,

$\sqrt{a x^{2} + b x + c} = p x + q$ $sqrt(ax^2+bx+c)=px+q$
$\sqrt{a x^{2} + b x + c} = \sqrt{a} x + \sqrt{c}$ $color(white)(sqrt(ax^2+bx+c))=sqrt a" "x +sqrt c$ ,

if

$a \geq 0$ $a>=0$ ,
$c \geq 0$ $c>=0$ , and
$b = \pm 2 \sqrt{a c}$ $b=+-2sqrt(ac)$ .

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What is the square root of ax^2+bx+c?

$\sqrt{a x^{2} + b x + c} = ?$ $sqrt(ax^2+bx+c)=?$ i tried myself and got $a^{2} x + \sqrt{c}, a^{2} x + \frac{b}{2}$ $a^(2)x+sqrt(c),a^(2)x+b/2$ But i do not think That is exactly right.

1 Answer

Explanation:

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What is the square root of ax^2+bx+c?

√ax2+bx+c=?sqrt(ax^2+bx+c)=? i tried myself and got a2x+√c,a2x+b2a^(2)x+sqrt(c),a^(2)x+b/2 But i do not think That is exactly right.

1 Answer

Explanation:

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Impact of this question

$\sqrt{a x^{2} + b x + c} = ?$ $sqrt(ax^2+bx+c)=?$ i tried myself and got $a^{2} x + \sqrt{c}, a^{2} x + \frac{b}{2}$ $a^(2)x+sqrt(c),a^(2)x+b/2$ But i do not think That is exactly right.