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2 years ago

PLEASE HELP I DONT UNDERSTAND

Mathematics

1 answer:

Charra [1.4K]2 years ago

8 0

Example Question:

X²+6x-14=0

Quadratic equations are usually in the form Ax²+Bx=C. Completing the Square is a technique for rewriting this equation in the form (x+a²)+b.

The example I provided is in standard form.

Here's what I did:

Step 1: Move the "C" to the other side (in this case it's negative 14, so you add 14 to both sides of the equation to move it).

Step 2: Leave a blank after Ax²+Bx. In this case, it'll look like: X²+6x+_____= 14.

Step 3: Use the box to find the missing "C". Idk if you can see the photo after the 1st one, but I circled the spaces in green. Those spaces indicate 2 factors that add up for B (in this case, 6). Those same factors will eventually multiply each other to get 9.

Step 4: Give the new "C" to both sides of the equation. What this means is that after you multiply the factors on the outside of the box, you put the number on both sides. The 3rd picture I drew is the process of finding the outer factors (by plugging in 3x to the spaces that were circled).

If you can't see it, it's (X+3)•(X+3).

The 4th picture is me plugging the new "C" (9) to both sides of the equation. 3 times 3 is 9. You multiply the 3's on the outside to get the factors on the inside.

Step 5: Use Step 3 box to write (perfect square)² of left hand side.

What this means is that you take the factors on the outside and rewrite it in perfect square form. like shown in picture 5. If you can't see it, it's like this:

(X+3)•(X+3) turns to (X+3)²=23.

The 23 was from adding 14+9 together.

So the final Answer would be:

(X+3)²=23.

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Implicit differentiation Please help

Anvisha [2.4K]

Answer:

$y''(-1) =8$

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

Brackets
Parenthesis
Exponents
Multiplication
Division
Addition
Subtraction

Left to Right

Equality Properties

Algebra I

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The derivative of a constant is equal to 0

Basic Power Rule:

f(x) = cxⁿ
f’(x) = c·nxⁿ⁻¹

Product Rule: $\frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)$

Chain Rule: $\frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)$

Quotient Rule: $\frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}$

Step-by-step explanation:

Step 1: Define

-xy - 2y = -4

Rate of change of the tangent line at point (-1, 4)

Step 2: Differentiate Pt. 1

Find 1st Derivative

Implicit Differentiation [Product Rule/Basic Power Rule]: $-y - xy' - 2y' = 0$
[Algebra] Isolate y' terms: $-xy' - 2y' = y$
[Algebra] Factor y': $y'(-x - 2) = y$
[Algebra] Isolate y': $y' = \frac{y}{-x-2}$
[Algebra] Rewrite: $y' = \frac{-y}{x+2}$

Step 3: Find y

Define equation: $-xy - 2y = -4$
Factor y: $y(-x - 2) = -4$
Isolate y: $y = \frac{-4}{-x-2}$
Simplify: $y = \frac{4}{x+2}$

Step 4: Rewrite 1st Derivative

[Algebra] Substitute in y: $y' = \frac{-\frac{4}{x+2} }{x+2}$
[Algebra] Simplify: $y' = \frac{-4}{(x+2)^2}$

Step 5: Differentiate Pt. 2

Find 2nd Derivative

Differentiate [Quotient Rule/Basic Power Rule]: $y'' = \frac{0(x+2)^2 - 8 \cdot 2(x + 2) \cdot 1}{[(x + 2)^2]^2}$
[Derivative] Simplify: $y'' = \frac{8}{(x+2)^3}$

Step 6: Find Slope at Given Point

[Algebra] Substitute in x: $y''(-1) = \frac{8}{(-1+2)^3}$
[Algebra] Evaluate: $y''(-1) =8$

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3 years ago

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can someone please help me figure this out? this is a summative and I'm failing for the year. it's algebra 1.

34kurt

Part (a)

x = number of bass caught
y = number of trout caught

x and y are nonnegative whole numbers.

3x represents the total weight of all the bass, since one bass weighs 3 pounds. Similarly, 2y is the total weight of the trout with each one being 2 pounds. Adding up all the weight of all the fish, we get 3x+2y. This is set equal to 34 pounds which is the total stated weight caught.

In short, we end up with the equation 3x+2y = 34

Now onto the second equation. The expression 5x represents how many points Pat gets from catching x bass, because he gets 5 points per bass. Then we subtract off 2y to represent Pat losing 2 points per trout he catches. Overall, his point total is 5x-2y. This is set equal to his point total of 14.

The second equation is 5x-2y = 14

------------

Answer:

3x+2y = 34

5x-2y = 14

Both equations are needed to form this particular system.

===================================================

Part (b)

We could use a number of methods to solve for (x,y). Possibly the most efficient is elimination. Why? Because note how we have 2y up top and -2y down below. They add to 0y which is just 0. The y terms go away, and are eliminated. Afterward, all we're left with is x in which we can solve for just like any other single variable equation. More info is in part (c).

===================================================

Part (c)

Refer to the answer in part (a). We'll add straight down. So that means we have these three sets of addition operations.

3x+5x becomes 8x
2y + (-2y), aka 2y-2y, becomes 0y and the y terms go away
34+14 simplifies to 48

After those three sets of addition operations, we're left with the much simpler equation 8x = 48. To solve this, we divide both sides by 8 and we get x = 6. This says Pat caught 6 bass.

Use this value of x to find y. So we'll plug x = 6 into either equation involving x,y and solve for y

3x+2y = 34

3(6)+2y = 34

18+2y = 34

2y = 34-18

2y = 16

y = 16/2

y = 8

So we can see that Pat caught 8 trout as well.

The check portion is shown in the attached image below. Effectively, all we're doing is replacing x and y with 6 and 8 respectively. Then we're simplifying both sides. Getting the same number on both sides results in a true equation, and therefore verifies we have the correct (x,y) solution.

Solution: (x,y) = (6,8)

Interpretation: Pat caught 6 bass and 8 trout

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3 years ago