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2 years ago

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Mathematics

1 answer:

8090 [49]2 years ago

6 0

9514 1404 393

Answer:

5x^2 +10x = -5

Step-by-step explanation:

The equation will have one solution when the discriminant is zero.

In standard form, the equation is ...

ax^2 +10x -c = 0

The discriminant of the equation ax^2+bx+c=0 is ...

b^2 -4ac

so the discriminant of your equation is ...

10^2 -4a(-c) = 100 +4ac

We want that to be zero, so we require ...

100 +4ac = 0

25 +ac = 0

ac = -25

Any pair of values of 'a' and 'c' that have a product of -25 will satisfy the requirement. As you state, one possible pair is a = 5 and c = -5. Putting these values into the given equation makes it ...

5x^2 +10x = -5 . . . . . the equation with a=5, c=-5

_____

Additional comment

Your problem statement does not require that 'a' be non-zero. If a=0, then the equation will have one real solution for any value of c.

If you restrict the solutions to integers, then possibilities include ...

(a, c) = (1, -25), (5, -5), (25, -1), (-25, 1), (-5, 5), (-1, 25)

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2 years ago

Read 2 more answers

Find the dot product of the position vectors whose terminal points are (14, 9) and (3, 6).

erma4kov [3.2K]

Answer:

Step-by-step explanation:

The formula for the dot product of vectors is

u·v = |u||v|cosθ

where |u| and |v| are the magnitudes (lengths) of the vectors. The formula for that is the same as Pythagorean's Theorem.

$|u|=\sqrt{14^2+9^2}$ which is $\sqrt{277}$

$|v|=\sqrt{3^2+6^2}$ which is $\sqrt{45}$

I am assuming by looking at the above that you can determine where the numbers under the square root signs came from. It's pretty apparent.

We also need the angle, which of course has its own formula.

$cos\theta=\frac{uv}{|u||v|}$ where uv has ITS own formula:

uv = (14 * 3) + (9 * 6) which is taking the numbers in the i positions in the first set of parenthesis and adding their product to the product of the numbers in the j positions.

uv = 96.

To get the denominator, multiply the lengths of the vectors together. Then take the inverse cosine of the whole mess:

$cos^{-1}\theta=\frac{96}{111.64676}$ which returns an angle measure of 30.7. Plugging that all into the dot product formula: