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3 years ago

Find the value of c such that y= x + c is a tangent to the parabola y = x squared - x - 12

Mathematics

1 answer:

Bond [772]3 years ago

5 0

Answer:

c = –13

Step-by-step explanation:

by property of tangent line, equation

x² – x – 12 = x + c

x² – 2x – 12 – c = 0

has only one solution, which mean

– 12 – c = 1

c = –13

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PLEASE HELP!! DUE SOON

mrs_skeptik [129]

Answer:

b=3a/8

a=8b/3

Step-by-step explanation:

b/a=k ( k is the constant proportional)

3/8=9/24=15/40

b/a=3/8

b=3a/8

a=8b/3

I hope this is right answer

7 0

3 years ago

Determine all prime numbers a, b and c for which the expression a ^ 2 + b ^ 2 + c ^ 2 - 1 is a perfect square .

kogti [31]

Answer:

The family of all prime numbers such that $a^{2} + b^{2} + c^{2} -1$ is a perfect square is represented by the following solution:

$a$ is an arbitrary prime number. (1)

$b = \sqrt{1 + 2\cdot a \cdot c}$ (2)

$c$ is another arbitrary prime number. (3)

Step-by-step explanation:

From Algebra we know that a second order polynomial is a perfect square if and only if $(x+y)^{2} = x^{2} + 2\cdot x\cdot y + y^{2}$ . From statement, we must fulfill the following identity:

$a^{2} + b^{2} + c^{2} - 1 = x^{2} + 2\cdot x\cdot y + y^{2}$

By Associative and Commutative properties, we can reorganize the expression as follows:

$a^{2} + (b^{2}-1) + c^{2} = x^{2} + 2\cdot x \cdot y + y^{2}$ (1)

Then, we have the following system of equations:

$x = a$ (2)

$(b^{2}-1) = 2\cdot x\cdot y$ (3)

$y = c$ (4)

By (2) and (4) in (3), we have the following expression:

$(b^{2} - 1) = 2\cdot a \cdot c$

$b^{2} = 1 + 2\cdot a \cdot c$

$b = \sqrt{1 + 2\cdot a\cdot c}$

From Number Theory, we remember that a number is prime if and only if is divisible both by 1 and by itself. Then, $a, b, c > 1$ . If $a$ , $b$ and $c$ are prime numbers, then $2\cdot a\cdot c$ must be an even composite number, which means that $a$ and $c$ can be either both odd numbers or a even number and a odd number. In the family of prime numbers, the only even number is 2.

In addition, $b$ must be a natural number, which means that:

$1 + 2\cdot a\cdot c \ge 4$

$2\cdot a \cdot c \ge 3$

$a\cdot c \ge \frac{3}{2}$

But the lowest possible product made by two prime numbers is $2^{2} = 4$ . Hence, $a\cdot c \ge 4$ .

The family of all prime numbers such that $a^{2} + b^{2} + c^{2} -1$ is a perfect square is represented by the following solution:

$a$ is an arbitrary prime number. (1)

$b = \sqrt{1 + 2\cdot a \cdot c}$ (2)

$c$ is another arbitrary prime number. (3)

Example: $a = 2$ , $c = 2$

$b = \sqrt{1 + 2\cdot (2)\cdot (2)}$

$b = 3$

4 0

3 years ago

The fence around a circular garden is 18 feet long. What is the area of the garden to the nearest square foot? Use 3.14 for pi.

Sidana [21]

Answer:

A = 26 ft²

Step-by-step explanation:

To find the area of a circle, use the formula A = πr²

We know π = 3.14 and C = 18.

Use the circumference to find "r". C = 2πr

C = 2πr

18 = 2(3.14)r

18 = 6.28r

r = 18/6.28 Exact answer

r ≈ 2.866 Rounded to three decimal places

You can use either of the two bolded "r" values. I will use the rounded answer in the formula for area.

Substitute π = 3.14 and r = 2.866.

A = πr²

A = (3.14)(2.866)² Solve exponents first

A = 3.14(8.213956) Multiply

A = 25.7918218... Unrounded answer

A ≈ 26 Rounded to nearest square foot

Add the units back in.

7 0

3 years ago

1 millimeter on the map represents 1 meter on the ground. What is the ratio

qwelly [4]

1mm:1m

since its just one to one, thats the ratio

3 0

3 years ago

How do i prove 2sec^2x-2sec^2xsin^2-sin^2x-cos^2x=1

kramer

$2\sec^2x-2\sec^2x\sin^2x-\sin^2x-\cos^2x=2\sec^2x(1-\sin^2x)-(\sin^2x+\cos^2x)$

Since $\sin^2x+\cos^2x=1$ , you have

$2\sec^2x(1-\sin^2x)-(\sin^2x+\cos^2x)=2\sec^2x\cos^2x-1$

and since $\sec x=\dfrac1{\cos x}$ , you end up with $2-1=1$ .

6 0

3 years ago