MATHEMATICS HIGH SCHOOL

Let f be the function defined by f(x) =x^4 -3x^2 +2A)find the zeros of f
B)write an equation of the line tangent to the graph of f at the point where x=1
C) find the x coordinate of each point at which the line tangent to the graph of f is parallel to the line y=-2x+4

Answers

Answer 1

Answer:

A convenient way to find the zeros of $f(x)=x^4-3x^2 +2$ is by factoring.

a) The equation,

$x^4-3x^2 +2=0$

can be rewritten as,

$(x^2)^2-3(x^2) +2=0$

We can think of this equation as a quadratic equation in $x^2$ , with $a=1,b=-3\:\: and \:\: c=2$ .

Observe that $ac=1 * 2=2$ .

We find two factors of 2 that adds up to $-3$ . These are, $-2,-1$ .

Now let us split the middle term. to obtain;

$(x^2)^2-(x^2) -2(x^2)+2=0$

We can factor to get,

$(x^2)(x^2-1)-2((x^2-1)=0$

We factor further to obtain;

$(x^2-1)((x^2-2)=0$

$\Rightarrow (x-1)(x+1((x- √(2))(x+ √(2))=0$

Hence the zeroes of $f(x)$ are;

$x=1,x=-1,x= √(2),x=- √(2)$

b) To find the line tangent, we must first, find the slope using differentiation. That is,

$Slope\:\: function=f'(x)=4x^3-6x$

At $x=1$ ,

$Slope=f'(1)=4(1)^3-6(1)=-2$

Also, we need to determine the $y$ value at $x=1$ . That is;

$f(1)=(1)^4-3(1)^2+2=0$

Now we can use the slope $m=-2$ and the point $(1,0)$ to write ythe equation of the line tangent.

$y-y_1 =m(x-x_1)$

$\Rightarrow y-0 =-2(x-1)$

$\Rightarrow y=-2x+2$

If the line tangent is parallel to the line $y=-2x+4$ , then

$f'(x)=-2$

Since parallel lines have the same slope.

$\Righarrow 4x^3-6x=-2$

$\Righarrow 4x^3-6x+2=0$

$\Righarrow (x-1)(2x- ( √(3)-1)(2x+ ( √(3)-1)=0$

Hence the x-coordinates are,

$x=1,x= ( √(3)-1)/(2),x= -(√(3)-1)/(2)$

Answer 2

Answer:
The zeros of the function can be determined by equating the equation to zero and determining the values of x.
0 = x4 - 3x2 + 2 the roots are x1 = sqrt 2 x2 = -sqrt 2x3 = 1x4 = -1
The tangent line is determined by differentiating the polynomial and substituting x by 1 to get the slope. That is, m = 4x^3 - 6 x = 4*1 - 6 = -2y-y1 = m(x-x1)when x =1 , y = 1-3+2 = 0
y - 0 = -2*(x-1)y = -2x + 2
c. y = -2x + 4 m = -2 -2 = 4x3 - 6x x= -1.3666 ; y =-0.1160 x= 1.3666 ; y =--0.1160
x =1 ; y = 0

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Solve the system of equations.2.5y + 3x = 27
5x – 2.5y = 5

What equation is the result of adding the two equations?

What is the solution to the system?

Answers

If you put them together, +2.5y and -2.5y neutralize themselves while 5x and 3x become 8x.

2.5y + 3x = 27
5x – 2.5y = 5

8x = 32
x = 4

2.5y + 3x4 = 27
2.5y + 12 = 27
2.5y = 27-12
2.5y = 15
y = 15/2.5
y = 6

X = 4
Y = 6

Answer:

x=4 Y=6

Step-by-step explanation:

Suppose S and T are mutually exclusive events. Find P(S or T). P(S) = 20%, P(T) = 22%

Answers

S and P are mutually exclusive events.
P ( S ) = 20% = 0.20
P ( T ) = 22 % = 0.22
The probability of mutually exclusive events:
P ( S or T ) = P ( S ) + P ( T )
P ( S or T ) = 0.20 +0.22 = 0.42
Answer: The probability of ( S or T ) is 0.42 or 42 %.

The probability of event S or event T is 42%.

What is the probability of event S or event T?

Mutually exclusive events are independent events. Independents events are events whose occurrence do not depend on each other.

The probability of event S or event T = 20 + 22 = 42%

To learn more about mutually exclusive events, please check: brainly.com/question/13075229

#SPJ5

X+2y=14
x-3y=-11
substitution

Answers

To solve this system of equations using substitution, we can solve one equation for one variable and substitute it into the other equation. Let's solve the first equation for x:

x = 14 - 2y

Now, substitute this expression for x in the second equation:

(14 - 2y) - 3y = -11

Simplify the equation:

14 - 2y - 3y = -11

Combine like terms:

14 - 5y = -11

Now, solve for y:

-5y = -11 - 14

-5y = -25

y = -25 / -5

y = 5

Now, substitute the value of y back into the first equation to solve for x:

x + 2(5) = 14

x + 10 = 14

x = 14 - 10

x = 4

So, the solution to the system of equations is x = 4 and y = 5.
Consider brainliesttt

You cannot bisect an angle using paper folding constructions True or False

Answers

Answer:

The given statement:

You cannot bisect an angle using paper folding constructions is a False Statement.

Step-by-step explanation:

The construction of a angle bisector with the help of paper folding is done by:

Draw a angle ACB on a piece of paper such that the ray CB lie on the base.

Fold the paper in such a way that ray AC falls along the ray CB.

draw the line along the crease that is formed on the paper starting from point C.

So, the ray so formed is the angle bisector of the angle ACB.

How many trapezoids can you find in this grid of 12-points using four points as the vertices of a trapezoid?

Answers

Answer:

Can I see the graph?

Step-by-step explanation:

A radio commercial for a loan company states: “You only pay $0.29 a day for. each $500 borrowed.” If you borrow $1,500 for 120 days, what amount will you. repay, and what annual interest rate is the company actually charging?

Answers

Total Interest will be = .29*1500*12 / 500 = 104.4
so total amount to be repay = 1500+104.4=1604.4
and annual interest rate = 104.4*100*360/1500*120=20.88%
hope it helps

Answer:

Total amount to be repay is $ 1604.4

The annual interest rate the company actually charging is 25.40%

Step-by-step explanation:

Given: A radio commercial for a loan company states: “You only pay $0.29 a day for each $500 borrowed.” We have borrowed $1,500 for 120 days.

We have to determine the amount to be repay and the annual interest rate the company actually charging.

UNITARY method is a method in which we first find the value of a single unit and then find the desired value by multiply it with unit value.

Since, we have to pay $0.29 a day for each $500 borrowed.

Thus, when we borrow $1 , we pay $(0.29)/(500)$