Question

solve x^2+9x=-9 if exact roots cannot be found. state the consecutive integers betweern which the roots are located.

Answer 1

Use the quadratic formula to find the two roots
\[
x= \frac{3}{2} \left(-3-\sqrt{5}\right)\\
x= \frac{3}{2}
\left(\sqrt{5}-3\right)
\]
These are exact. May be they mean by exact like rationals.

Answer 2

The first root is between -8 and -7
the second -2 and -1

Answer 3

Do you know anything about quadratics?

\[x^2+9x=-9\]\[x^2+9x+9=0\]That's a parabola which opens up, and has a vertex at \(x=-\frac{9}{2}\). We know this because it is in the form \[y=ax^2+bx+c\]with \(a>0\) (which means it opens upward) and the vertex appears at \(x=-\frac{b}{2a}\)

The roots of the equation are simply the points at which it crosses the x-axis, or if you prefer, intersects the line \(y = 0\). A parabola is symmetric about the vertex, so we know the roots will be at \(x=-\frac{9}{2}\pm k\) where \(k\) is some constant. We can bracket the roots by evaluating the function at various values of \(x\) and observing whether the result changes sign. If \(f(a) > 0\) and \(f(b) < 0\) then there must be a root between \(x=a\) and \(x=b\) if the function is continuous (which this is).

Answer 4

@whpalmer4 thank you so much!
I honestly don't know much because the teacher I have really doesnt teach us thank you! :)

Answer 5

If we make a table of values around the vertex, we can see where the roots must be:

\[\begin{array}{cc}
x & y \\ -4.5 & -11.25 \\
-5 & -11 \\
-6 & -9 \\
-7 & -5 \\
-8 & 1 \\
-9 & 9 \\
\end{array}\]
Clearly there must be a root between \(x=-7\) and \(x=-8\)

The parabola is symmetric, as I mentioned, so the other root would be the two integers just as far on the other side of -4.5

Answer 6

if a teacher is not teaching well, just grab a book. There are many books that will read very easy.

Answer 7

-7-(-4.5) = -7+4.5 = -2.5
-4.5-(-2.5) = -4.5+2.5 = -2
So the other root is between x = -2 and x = -1, as @eliassaab said

Here's a graph:

(the vertical line indicates the vertex)

Answer 8

A little thought should reveal that we could use this technique (looking for the sign changes) to narrow in ever closer on the actual root, if some reason we couldn't find it directly. We could "chop" the distance between \(x =-8\) and \(x=-7\) into little pieces and repeat the process there:

\[\begin{array}{cc}
-8. & 1. \\
-7.9 & 0.31 \\
-7.8 & -0.36 \\
-7.7 & -1.01 \\
-7.6 & -1.64 \\
-7.5 & -2.25 \\
-7.4 & -2.84 \\
-7.3 & -3.41 \\
-7.2 & -3.96 \\
-7.1 & -4.49 \\
-7. & -5. \\
\end{array}\]
Clearly, the root is between \(x = -7.9\) and \(x=-7.8\)

Let's do that again:
\[\begin{array}{cc}
-7.9 & 0.31 \\
-7.89 & 0.2421 \\
-7.88 & 0.1744 \\
-7.87 & 0.1069 \\
-7.86 & 0.0396 \\
-7.85 & -0.0275 \\
-7.84 & -0.0944 \\
-7.83 & -0.1611 \\
-7.82 & -0.2276 \\
-7.81 & -0.2939 \\
-7.8 & -0.36 \\
\end{array}\]
Okay, now we see it is between \(x = -7.86\) and \(x=-7.85\). The approximate value if you evaluate the square roots above is \(x\approx -7.8541\)

Answer 9

There are methods which let you make better guesses for the next attempt than this, but this clearly shows how you could do it.

Answer 10

@zzr0ck3r trust me ive been looking for books for the past weeks and yet none have really helped me :)

Answer 11

@leogarcia13 have you tried https://www.khanacademy.org/ ?

Answer 12

I've generally enjoyed the Khan Academy videos I've watched on a number of topics, and I know there are a lot of them about mathematics. You might check them out. https://www.khanacademy.org/math/algebra/quadratics

Answer 13

lol jinx

Answer 14

@robtobey sometimes I think I help out on OpenStudy just to have more excuses to make plots with Mathematica :-)

Answer 15

this is the first time i ever hear about khanacadamey.org well thank you too for the info its going to help me out even much more @zzr0ck3r @whpalmer4

Answer 16

@whpalmer4

No plots today. Unable to post.