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lara [203]
3 years ago
7

When the solutions are going to be complex, which methods can be used to solve quadratic equations?

Mathematics
1 answer:
yKpoI14uk [10]3 years ago
7 0
You could complete the square to state the vertex. 
You could use the quadratic equation to find the roots (which are complex). 

Try an example that will require both.

y = x^2 + 2x + 5

Step One
Get the graph. That's included below.

Step Two
Provide the steps for completing the square.
Note: we should get (-1,4)

y= (x^2 +2x ) + 5
y = (x^2 +2x + 1) + 5 - 1
y = (x +1)^2 + 4

The vertex is at (-1,4)

Step Three
Find the roots. Use the quadratic equation. Note that the graph shows us that the equation never crosses or touches the x axis. The roots are complex.

\text{x = }\dfrac{ -b \pm \sqrt{b^{2} - 4ac } }{2a}

a = 1
b = 2
c = 5

\text{x = }\dfrac{ -2 \pm \sqrt{2^{2} - 4*1*5 } }{2}
\text{x = }\dfrac{ -2 \pm \sqrt{4 - 20 } }{2}
\text{x = }\dfrac{ -2 \pm \sqrt{-16 } }{2}
\text{x = }\dfrac{ -2 \pm 4i }{2}
x = -1 +/- 2i

x1 = -1 + 2i
x2 = -1 - 2i And we are done.


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What are the zeros of f(x)=x(x-8)
Flura [38]

Answer:

The zeros are x=0,x=8

Step-by-step explanation:

f(x)=x(x-8)

To find the zeros, we set the equation equal to zero

0 = x(x-8)

Using the zero product property

x= 0  x-8=0

x=0  x=8

The zeros are 0,8

8 0
3 years ago
Read 2 more answers
Evaluate each finite series for the specified number of terms. 1+2+4+...;n=5
zaharov [31]

Answer:

31

Step-by-step explanation:

The series are given as geometric series because these terms have common ratio and not common difference.

Our common ratio is 2 because:

1*2 = 2

2*2 = 4

The summation formula for geometric series (r ≠ 1) is:

\displaystyle \large{S_n=\frac{a_1(r^n-1)}{r-1}} or \displaystyle \large{S_n=\frac{a_1(1-r^n)}{1-r}}

You may use either one of these formulas but I’ll use the first formula.

We are also given that n = 5, meaning we are adding up 5 terms in the series, substitute n = 5 in along with r = 2 and first term = 1.

\displaystyle \large{S_5=\frac{1(2^5-1)}{2-1}}\\\displaystyle \large{S_5=\frac{2^5-1}{1}}\\\displaystyle \large{S_5=2^5-1}\\\displaystyle \large{S_5=32-1}\\\displaystyle \large{S_5=31}

Therefore, the solution is 31.

__________________________________________________________

Summary

If the sequence has common ratio then the sequence or series is classified as geometric sequence/series.

Common Ratio can be found by either multiplying terms with common ratio to get the exact next sequence which can be expressed as \displaystyle \large{a_{n-1} \cdot r = a_n} meaning “previous term times ratio = next term” or you can also get the next term to divide with previous term which can be expressed as:

\displaystyle \large{r=\frac{a_{n+1}}{a_n}}

Once knowing which sequence or series is it, apply an appropriate formula for the series. For geometric series, apply the following three formulas:

\displaystyle \large{S_n=\frac{a_1(r^n-1)}{r-1}}\\\displaystyle \large{S_n=\frac{a_1(1-r^n)}{1-r}}

Above should be applied for series that have common ratio not equal to 1.

\displaystyle \large{S_n=a_1 \cdot n}

Above should be applied for series that have common ratio exactly equal to 1.

__________________________________________________________

Topics

Sequence & Series — Geometric Series

__________________________________________________________

Others

Let me know if you have any doubts about my answer, explanation or this question through comment!

__________________________________________________________

7 0
3 years ago
Find the range of the data set : -5, -6, -1, -3, 6, 10
Furkat [3]
<h3>Answer:  16</h3>

Work Shown:

range = max - min = 10 - (-6) = 10 + 6 = 16

The range is the distance from the smallest item to the largest. Use of a number line might help sort out the values to see which is the smallest and largest.

5 0
2 years ago
Heather is cashier, she can ring up 12 customers in 9 minutes at this rate how many minutes does it take to ring up 4 customers
stellarik [79]

Answer:

5.2 minutes for 4 costomers.

Step-by-step explanation:

12/9=1.333 (3 repeating)

1.3 x 4 = 5.2

5 0
3 years ago
Which completely describes the polygon ? equilateral equiangular regular congruent
VladimirAG [237]
<h3>Answer: C) regular</h3>

Explanation:

Any regular polygon is both equilateral and equiangular.

Equilateral = all sides are equal

Equiangular = all angles are equal

5 0
3 years ago
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