Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Given an equation $6 + 4x - 2x^2 = 0$ . (a) Express $6 + 4x - 2x^2$ in the form $a - (x + b)^2$

View step-by-step help

View step-by-step help

Find equations of tangent lines using limits

Full solution

Q. Given an equation

6 + 4x - 2x^2 = 0

. (a) Express

6 + 4x - 2x^2

in the form

a - (x + b)^2

Arrange in Descending Order: We need to complete the square to express the given quadratic equation in the form $a - (x + b)^2$ . The first step is to arrange the equation in descending order of the powers of $x$ . $\newline$ $6 + 4x - 2x^2$ rearranged is $-2x^2 + 4x + 6$ .
Factor Out Coefficient: Next, we factor out the coefficient of the $x^2$ term from the $x^2$ and $x$ terms. $\newline$ This gives us $-2(x^2 - 2x) + 6$ .
Find Value of b: Now, we need to find the value of $b$ in the expression $(x + b)^2$ that will complete the square. The term $b$ is half of the coefficient of $x$ , which is $-2$ in this case. $\newline$ So, $b = -2 / 2 = -1$ .
Add/Subtract $b^2$ : We add and subtract $b^2$ inside the parentheses to complete the square. Since $b = -1$ , $b^2 = (-1)^2 = 1$ . We add and subtract $1$ inside the parentheses, which gives us $-2(x^2 - 2x + 1 - 1) + 6$ .
Write as Perfect Square: Now we can write the expression as a perfect square: $-2((x - 1)^2 - 1) + 6$ .
Distribute $-2$ : Next, we distribute the $-2$ inside the parentheses: $-2(x - 1)^2 + 2 + 6$ .
Combine Constant Terms: Finally, we combine the constant terms: $-2(x - 1)^2 + 8$ .

More problems from Find equations of tangent lines using limits

Question

Consider the curve given by the equation

x y^{2}+5 x y=50

. It can be shown that

\frac{d y}{d x}=\frac{-y(y+5)}{x(2 y+5)} \text {. }

\newline

Write the equation of the vertical line that is tangent to the curve.

Posted 1 month ago

Question

The rate of change

\frac{d P}{d t}

of the number of algae in a tank is modeled by the following differential equation:

\newline

\frac{d P}{d t}=\frac{2317}{10614} P\left(1-\frac{P}{662}\right)

\newline

t=0

, the number of algae in the tank is

174

and is increasing at a rate of

28

algae per minute. At what value of

P

P(t)

growing the fastest?

\newline

Posted 2 months ago

Question

The rate of change

\frac{d P}{d t}

of the number of bacteria in a tank is modeled by the following differential equation:

\newline

\frac{d P}{d t}=\frac{2}{9849} P(598-P)

\newline

t=0

, the number of bacteria in the tank is

196

and is increasing at a rate of

16

bacteria per minute. At what value of

P

does the graph of

P(t)

have an inflection point?

\newline

Posted 2 months ago

Question

The rate of change

\frac{d P}{d t}

of the number of people infected by a disease is modeled by the following differential equation:

\newline

\frac{d P}{d t}=\frac{45}{125404} P(800-P)

\newline

t=0

, the number of people infected by the disease is

214

and is increasing at a rate of

45

people per hour. What is the limiting value for the total number of people infected by the disease as time increases?

\newline

Posted 2 months ago

Question

Which of the following is a correct interpretation of the expression

\newline

-4+13

\newline

1

\newline

(A) The number that is

4

-13

on the number line

\newline

(B) The number that is

4

to the right of

-13

on the number line

\newline

(C) The number that is

13

-4

on the number line

\newline

(D) The number that is

13

to the right of

-4

on the number line

Posted 2 months ago

Question

f

f(x)=x^{3}+2 \sin (3 x-3)

. Use a calculator to write the equation of the line tangent to the graph of

f

x=0.5

. You should round all decimals to

3

\newline

Posted 2 months ago

Question

f

f(x)=x^{2}-2 x+3 \cos \left(x^{2}-x\right)

. Use a calculator to write the equation of the line tangent to the graph of

f

x=-2.5

. You should round all decimals to

3

\newline

Posted 2 months ago

Question

f

f(x)=x^{2}+x-2 \sin (2 x)

. Use a calculator to write the equation of the line tangent to the graph of

f

x=3

. You should round all decimals to

3

\newline

Posted 2 months ago

Question

f

f(x)=x^{3}+\cos \left(2 x^{2}+5\right)

. Use a calculator to write the equation of the line tangent to the graph of

f

x=1

. You should round all decimals to

3

\newline

Posted 2 months ago

Question

f

f(x)=x^{3}+3-5 \sin \left(x^{2}\right)

. Use a calculator to write the equation of the line tangent to the graph of

f

x=1

. You should round all decimals to

3

\newline

Posted 2 months ago