Determine the real roots of fx 06 x224 x55 a graphically b...

Determine the real roots of $f(x)=-0.6 x^{2}+2.4 x+5.5:$ (a) Graphically. (b) Using the quadratic formula. (c) Using three iterations of the bisection method to determine the highest root. Employ initial guesses of $x_{1}=5$ and $x_{u}=10$ Compute the estimated error $\varepsilon_{a}$ and the true error $\varepsilon_{t}$ after each iteration.

Transcript

00:01 So in this problem, we are asked to determine the real roots, the real roots, for this polynomial.

00:17 F of x equals minus 0 .6x squared plus 2 .4x plus 2 .4x plus 5 .5.

00:34 Okay, so first of all, we're asked to do it graphically.

00:37 So let's just go over to our graphing calculator real quick.

00:40 And enter the formula minus 0 .6 squared plus 2 .4x plus 5 .5.

01:02 And i needed to say if you're the front that y equals that.

01:16 And there is our graph right there.

01:22 And so where do we have the roots at? first root will be right here, right? at minus 1 .629.

01:35 So first of all, i asked to do graphically, at minus 1 .629 and 5 .629.

01:53 5 .629, or 2 .9.

01:57 Or 2 roots.

01:59 Okay? next, we're asked to do this with the quadratic formula.

02:04 Remember the quadratic formula.

02:07 Minus b plus or minus a square root b squared minus 4 a c all over 2a well we got to to determine a b and c then don't we okay and what we notice about our formula up here for a second we notice that first of all everything has a one decimal place in it right a tenth decimal place holder in it so we can multiply everything by ten and still be whole and we could multiply everything everything by negative to get rid of this negative up front.

02:46 Okay? so when we do that, what do we have? we have f of x equals 0.

02:53 So it equals 6x squared minus 24x minus 55 because we multiply it through by the negative.

03:07 And so that gives us then that a is 6, b is 2.

03:15 24 with a negative on it and c is negative 55 all right so using the quadratic formula x equals minus b well minus a negative 24 that's a positive 24 plus or minus square root of minus 24 minus 24 squared minus four times a is six c is minus 55 all over two times two times two times times a which is six okay so this now this now is me uh i'll put that equals there i want to do this instead is 24 plus or minus a square root of well let's see what's negative 24 squared 576 a minus times a minus is plus 4 times 6 that's 1320 all over 12.

04:46 Okay.

04:47 So this is now 24 plus or minus.

04:54 Now let's see.

04:55 1320 plus 576 is the square root of 1896 all over 12.

05:09 And so this is now 24 plus the square root of 1896 all over 12, which is square root of 1896 plus 24 divided by 12.

05:44 This is 5 .629.

05:52 And i got 24 minus the square root of 1896.

05:58 All over 12, which is a negative 1 .629.

06:26 All right, so we've got the same answers, which is what we should have gotten.

06:30 Okay? so then part c asks us to use three iterations of the bisection method, determine the highest root...