Consider the equation 5x^2-10x+c=0. What values of c result in the equation having a complex solutions? Represent your answer on
the number line
1 answer:
Answer:
When we have a quadratic equation:
a*x^2 + b*x + c = 0
There is something called the determinant, and this is:
D = b^2 - 4*a*c
If D < 0, then the we will have complex solutions.
In our case, we have
5*x^2 - 10*x + c = 0
Then the determinant is:
D = (-10)^2 - 4*5*c = 100 - 4*5*c
And we want this to be smaller than zero, then let's find the value of c such that the determinant is exactly zero:
D = 0 = 100 - 4*5*c
4*5*c = 100
20*c = 100
c = 100/20 = 5
As c is multiplicating the negative term in the equation, if c increases, then we will have that D < 0.
This means that c must be larger than 5 if we want to have complex solutions,
c > 5.
I can not represent this in your number line, but this would be represented with a white dot in the five, that extends infinitely to the right, something like the image below:
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Answer:
(a)
(b) The graph is shown below.
(c) Seiji earned $ 210 and Gavin earned $ 240.
Step-by-step explanation:
Given:
Sum of earnings of Seiji and Gavin = $ 450
Gavin earns $ 30 more that that of Seiji.
(a)
So, as per question, if s and g are the amounts earned by Seiji and Gavin respectively, then;
(b)
The graph is plotted using the x and y intercepts of each line.
For the line, , the x intercept is at
. So, x intercept is (450,0). Similarly, the y intercept is when s=0, which is (0,450). Draw a line passing through these two points. The red line represents
.
We follow the same process to plot the next line. The x intercept is at (-30,0) and y intercept is at (0,30). Draw a line passing through these two points. The blue line represents .
The graph is shown below.
(c)
From the graph, the point of intersection of the two lines gives the earnings of each of the persons.
The value of the point of intersection is the earning of Seiji which is $ 210 and the
value is the earning of Gavin which is $ 240 as shown in the graph.
Therefore, Seiji earned $ 210 and Gavin earned $ 240.
We need to account for both x values on either side of the length, and width.
Thus, the length becomes 10 + x + x = 10 + 2x
and the width becomes 5 + x + x = 5 + 2x
For the second question, I'm assuming we don't account for the area that is covered by the garden.
Then we can say that the path is measured by: (5 + 2x)(10 + 2x) - 50, which is the area of the garden itself.
(5 + 2x)(10 + 2x) - 50 = 54
Expanding the brackets:
x = -9, or x = 3/2
Since x > 0, then x ≠ -9
Thus, the only x-value we can take is x = 3/2
Thus, the length becomes 10 + x + x = 10 + 2x
and the width becomes 5 + x + x = 5 + 2x
For the second question, I'm assuming we don't account for the area that is covered by the garden.
Then we can say that the path is measured by: (5 + 2x)(10 + 2x) - 50, which is the area of the garden itself.
(5 + 2x)(10 + 2x) - 50 = 54
Expanding the brackets:
x = -9, or x = 3/2
Since x > 0, then x ≠ -9
Thus, the only x-value we can take is x = 3/2