t²-32t - 2336=0
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Answers
Answer:
The first term is, t2 its coefficient is 1 .
The middle term is, -32t its coefficient is -32 .
The last term, "the constant", is -160
Step-1 : Multiply the coefficient of the first term by the constant 1 • -160 = -160
Step-2 : Find two factors of -160 whose sum equals the coefficient of the middle term, which is -32 .
-160 + 1 = -159
-80 + 2 = -78
-40 + 4 = -36
-32 + 5 = -27
-20 + 8 = -12
-16 + 10 = -6
-10 + 16 = 6
-8 + 20 = 12
-5 + 32 = 27
-4 + 40 = 36
-2 + 80 = 78
-1 + 160 = 159
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Equation at the end of step
1
:
t2 - 32t - 160 = 0
STEP
2
:
Parabola, Finding the Vertex
2.1 Find the Vertex of y = t2-32t-160
Parabolas have a highest or a lowest point called the Vertex . Our parabola opens up and accordingly has a lowest point (AKA absolute minimum) . We know this even before plotting "y" because the coefficient of the first term, 1 , is positive (greater than zero).
Each parabola has a vertical line of symmetry that passes through its vertex. Because of this symmetry, the line of symmetry would, for example, pass through the midpoint of the two x -intercepts (roots or solutions) of the parabola. That is, if the parabola has indeed two real solutions.
Parabolas can model many real life situations, such as the height above ground, of an object thrown upward, after some period of time. The vertex of the parabola can provide us with information, such as the maximum height that object, thrown upwards, can reach. For this reason we want to be able to find the coordinates of the vertex.
For any parabola,At2+Bt+C,the t -coordinate of the vertex is given by -B/(2A) . In our case the t coordinate is 16.0000
Plugging into the parabola formula 16.0000 for t we can calculate the y -coordinate :
y = 1.0 * 16.00 * 16.00 - 32.0 * 16.00 - 160.0
or y = -416.000
Parabola, Graphing Vertex and X-Intercepts :
Root plot for : y = t2-32t-160
Axis of Symmetry (dashed) {t}={16.00}
Vertex at {t,y} = {16.00,-416.00}
t -Intercepts (Roots) :
Root 1 at {t,y} = {-4.40, 0.00}
Root 2 at {t,y} = {36.40, 0.00}
Solve Quadratic Equation by Completing The Square
2.2 Solving t2-32t-160 = 0 by Completing The Square .
Add 160 to both side of the equation :
t2-32t = 160
Now the clever bit: Take the coefficient of t , which is 32 , divide by two, giving 16 , and finally square it giving 256
Add 256 to both sides of the equation :
On the right hand side we have :
160 + 256 or, (160/1)+(256/1)
The common denominator of the two fractions is 1 Adding (160/1)+(256/1) gives 416/1
So adding to both sides we finally get :
t2-32t+256 = 416
Adding 256 has completed the left hand side into a perfect square :
t2-32t+256 =
(t-16) • (t-16) =
(t-16)2
Things which are equal to the same thing are also equal to one another. Since
t2-32t+256 = 416 and
t2-32t+256 = (t-16)2
then, according to the law of transitivity,
(t-16)2 = 416
We'll refer to this Equation as Eq. #2.2.1
The Square Root Principle says that When two things are equal, their square roots are equal.
Note that the square root of
(t-16)2 is
(t-16)2/2 =
(t-16)1 =
t-16
Now, applying the Square Root Principle to Eq. #2.2.1 we get:
t-16 = √ 416
Add 16 to both sides to obtain:
t = 16 + √ 416
Since a square root has two values, one positive and the other negative
t2 - 32t - 160 = 0
has two solutions:
t = 16 + √ 416
or
t = 16 - √ 416
Solve Quadratic Equation using the Quadratic Formula
2.3 Solving t2-32t-160 = 0 by the Quadratic Formula .
According to the Quadratic Formula, t , the solution for At2+Bt+C = 0 , where A, B and C are numbers, often called coefficients, is given by :
- B ± √ B2-4AC
t = ————————
2A
In our case, A = 1
B = -32
C = -160
Accordingly, B2 - 4AC =
1024 - (-640) =
1664
Applying the quadratic formula :
32 ± √ 1664
t = ——————
2
Can √ 1664 be simplified ?
Yes! The prime factorization of 1664 is
2•2•2•2•2•2•2•13
To be able to remove something from under the radical, there have to be 2 instances of it (because we are taking a square i.e. second root).
√ 1664 = √ 2•2•2•2•2•2•2•13 =2•2•2•√ 26 =
± 8 • √ 26
√ 26 , rounded to 4 decimal digits, is 5.0990
So now we are looking at:
t = ( 32 ± 8 • 5.099 ) / 2
Two real solutions:
t =(32+√1664)/2=16+4√ 26 = 36.396
or:
t =(32-√1664)/2=16-4√ 26 = -4.396
Two solutions were found :
t =(32-√1664)/2=16-4√ 26 = -4.396
t =(32+√1664)/2=16+4√ 26 = 36.396