Solve x2 + 4x + 12 = 0 by completing the square.

x = –2 + 2i, x = –2 –2i

x = –2 + i, x = –2 – i

x = –2 + 2i, x = –2 – 2i

x = 0, –4

Answer:

Short leg: 9 in

Long leg: 40 in

Hypotenuse: 41 in

Step-by-step explanation:

This problem involves the Pythagorean Theorem, solving a system of equations, and solving a quadratic equation.

The Pythagorean Theorem applies because the question states this is a right triangle.  That is one equation.  

Two more equations come from descriptive relationships between various sides of the triangle.

Solving the system is done here with the Substitution Method, and solving the quadratic equation is done with factoring and the zero product property.  Other methods could be used, but there is a character limit, so we can't dive into every way to solve this.

Pythagorean Theorem: [tex]a^2+b^2=c^2[/tex] where "c" is the hypotenuse of the right triangle.

"c" must be hypotenuse.

It doesn't matter which of the two legs' lengths is used for "a" & "b".  However, because of other parts of the question, we'll need to know which one is which.

The question refers back multiple times to one specific leg (the short one).  So, when we refer back to that leg in our equations, we need to make sure we're using the same leg each time.

Keeping track of which leg is which

Let's use "a" for the smaller leg, and "b" for the longer leg.

"c", as always, is the hypotenuse.

Next, the question gives two statements relating the side lengths of the triangle to each other.

For the first statement, "the length of the longer leg of a right triangle" (that's "b") "...is 13 inches more than three times the length of the shorter leg ("a").

Translating the first sentence into math:

[tex]b=3a+13[/tex]

Similarly from the second statement, we get:  [tex]c=3a+14[/tex]

Combining this with the Pythagorean Theorem, we have a system of 3 equations, and 3 unknowns.

[tex]b=3a+13[/tex]

[tex]c=3a+14[/tex]

[tex]a^2+b^2=c^2[/tex]

To solve a system of equations, there are two main methods:  

Either method can be used, but the substitution method is the most intuitive.

Substitution method

To solve a system with the substitution method, isolate variable in an equation, written in terms of the other variables, and substitute them in to try to get a single equation with a single unknown.

Looking at our system, the first two equations already have "b" and "c" isolated in terms of "a".  If we substitute both of these into the third equation, we'll have a single equation in terms of a, and can attempt to solve it.

[tex]b=3a+13[/tex]

[tex]c=3a+14[/tex]

[tex]a^2+b^2=c^2[/tex]

Substituting...

[tex]a^2+(3a+13)^2=(3a+14)^2[/tex]

Rewrite the square binomials as a product...

[tex]a^2+(3a+13)(3a+13)=(3a+14)(3a+14)[/tex]

Apply FOIL (a combination of the distributive property, the commutative properties of multiplication, and combining like terms)...

[tex]a^2+(9a^2+39a+39a+169)=(9a^2+42a+42a+196)\\a^2+(9a^2+78a+169)=(9a^2+84a+196)\\[/tex]

Observe that the equation is a degree-2 polynomial (polynomial with highest power of 2; aka quadratic).  Move all terms to one side to get the equation equal to zero in preparation for solving:

[tex]a^2+(9a^2+78a+169)=(9a^2+84a+196)\\a^2+9a^2+78a+169-(9a^2+84a+196)=0\\a^2-6a-27=0\\[/tex]

There are a few ways to solve this, including the quadratic formula.  However, factoring the polynomial and using the zero product property is quite efficient here:

Factoring to solve a Quadratic

Since the leading coefficient is 1, we can use the shortcut and find factors of -27 that add to make -6.

-27 has factor pairs of:

3 and -9 add to make -6, so factoring the polynomial is

[tex]a^2-6a-27=0\\(a+3)(a-9)=0[/tex]

Applying the zero product property...

[tex](a+3)=0[/tex] or [tex](a-9)=0[/tex]

So a=-3 or a=9.  Given that this represents a length in a triangle, we discard the negative solution.  So a=9.

Finding the other two unknowns

To finish up, substitute the "a" back into the two equations for "b" and "c".

[tex]b=3a+13\\b=3(9)+13\\b=27+13\\b=40[/tex]

[tex]c=3a+14\\c=3(9)+14\\c=27+14\\c=41[/tex]

The original problem set up our measurements in inches, so all three values are measured in inches.

Remember which length was which

"a": short leg

"b": longer leg

"c": hypotenuse

Short leg: 9 in

Long leg: 40 in

Hypotenuse: 41 in.

Answer:

252-232=20-1=19

Step-by-step explanation:

the question said in between so you don't count the first and the last numbers

Answer:

1/3

Step-by-step explanation:

Answer:

C

Step-by-step explanation:

y = 3x - 4

4y = 12x - 4

divide the bottom equation by 4

y = 3x - 4

y = 3x - 1

y = -3

3x - 1 = -3

3x = -2

x = -2/3

(-2/3, -3) is the answer. by this logic, there is a splution so can't be a, the answer is not (0, 4) so not b so it must be c

Answer:

< $0 --------------$35------ $50>

Step-by-step explanation:

wait that's not the full question but uh I assume that's 30% off $50?

it would be 0.3x50 which is 15 and 50-15 is 35.

the range is 0 - $50 if I'm not mistaken
< $0 -------------------------- $50> scuffed number line
but 35 is somewhere around here?
< $0 --------------$35------ $50>

Answer:

L1 and L2 are parallel

BAC = 58° as it is given in the question that AB bisects DAC

similarly BCE = y°

angle B = 90° ( from figure)

Y + 58° + 90° = 180 ( angle Sum property of a triangle).

Y + 148° = 180°

Y = 180 - 148

Y = 32°

After factorizing the given equation, we can find the three iterations by putting the consecutive values of x.

The given equation is 3∧x = 2∧(-x) + 4. We can modify it as shown below:

Take logarithm on both the sides

log(3∧x) = log[2∧(-x) + 4]

xlog3 = log[2∧(-x) + 4]

x = log[2∧(-x) + 4] / log3

For the first iteration, put x = 1

x = log[2∧(-1) + 4] / log3

x = log4.5 / log3

x = 0.477

For the second iteration, put x = 2

x = log[2∧(-2) + 4] / log3

x = log4.25 / log3

x = 0.477

For the third iteration, put x = 3

x = log[2∧(-3) + 4] / log3

x = log4.125 / log3

x = 0.477

So, this is how we can find three iterations.

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Step-by-step explanation:

b²+7b+10

Common factors are 5,2

b²+7b+10

b²+2b+5b+10

b(b+2)+5(b+2)

(b+2)(b+5)

Hope it helps.✨✨

The area of a 2D form is the amount of space within its perimeter. The surface area of the prism is 5x²+(24/5x)+(24/x).

The area of a 2D form is the amount of space within its perimeter. It is measured in square units such as cm2, m2, and so on. To find the area of a square formula or another quadrilateral, multiply its length by its width.

Let the width of the rectangular box be x. Therefore, the length of the box is 5x. Now, the height of the box can be written as,

Volume of the box = length × width × Height

12 = 5x² × H

H= 12 / 5x²

Further, the surface area of the open box is,

Surface area of the box

[tex]= (5x \times x) + 2(h \times x) + 2(5x \times h)\\\\= (5x^2) + 2(\dfrac{12}{5x^2} \times x) + 2(5x \times \dfrac{12}{5x^2})\\\\\\= 5x^2 + \dfrac{24}{5x} + \dfrac{24}{x}[/tex]

Hence, the surface area of the prism is 5x²+(24/5x)+(24/x).

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Answer:

Step-by-step explanation:

Solution,

We use the slope formula, setting x=a :

[tex] \displaystyle{{{{ \lim_{h \to0}}}} \frac{ \frac{2}{x + h} - \frac{2}{x} }{h} = \lim_{h \to0} \frac{ \frac{2}{a + h} - \frac{2}{a} }{h} }[/tex]

[tex] \displaystyle{ \lim_{h \to0} \frac{ \frac{2a - 2(a + h)}{a(a + h)} }{h} = \displaystyle{ \lim_{h \to0} \frac{2a - 2(a + h)}{ha(a + h)} }}[/tex]

[tex] \displaystyle{ \lim_{h \to0} \frac{2a - 2a - 2h}{ha(a + h)} = \displaystyle{ \lim_{h \to0} \frac{ - 2h}{ha(a + h)} }}[/tex]

[tex] \displaystyle{ \lim_{h \to0} \frac{ - 2}{a(a + h)} = - \frac{ 2}{ {a}^{2} } }[/tex]

Therefore, The slope of the curve y=2/x at x=a is -2/a^2.

Answer:

6,3

Step-by-step explanation:

hope this helps

the math section in part B shows the equation for the area of a triangle, but this math could be skipped with a similar problem if you just divide the parraellogram by 2. (logic section)

Answer:

A maybe?

Step-by-step explanation:

by looking at the graph you can see there are two points with one being at (0, 3) (given) and the other being at (1, 1) (by looking at the graph). You can use the slope formula (y2 - y1) / (x2 - x1) which gives you the slope.

Plug Values In:

(1-3) / (1-0) = -2/1 = -2

slope of f(x) = -2

slope of g(x) = -7 (slope-intercept form y=mx+b where m is the slope and b is the y-intercept)

I'm honestly not quite sure what it means by greater than since technically the slope of f(x) is greater than the slope of g(x) but does it mean how much it's changing by in which case the sign doesn't matter. Although I'm just going to assume it purely means the value in which case A is true.

Answer:

it is 18 by unknown number

Answer: 180 degrees

Step-by-step explanation:

The measure of the arc of a semicircle is 180 degrees.

Answer: 80%

Step-by-step explanation:

Since only 2 out of 10 women dropped out, 10 - 2 = 8

One woman = 10%, 8 x 10% = 80%

The percentage of the women who moved to the next stage is 80%.

Percentage can be described as a fraction out of an amount that is usually expressed as a number out of hundred.

Women who moved to the next stage: 10 - 2 = 8

(8/10) x 100 = 80%

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Answer: 46.0 ft

Step-by-step explanation:

[tex]\sin 24^{\circ}=\frac{x}{105}\\\\x=105\sin 24^{\circ}[/tex]

So, the distance above the ground is [tex]105\sin 24^{\circ}+3.25 \approx \boxed{46.0 \text{ ft}}[/tex]

Answer:

option b, y = 3/4x - 11

Step-by-step explanation:

Parallel lines have the same slope,

slope = rise / run where the rise is 3 and the run is 4

slope = 3/4

The line is in a upwards trend making the slope positive making choice c and d incorrect

The line passes through (8, -5)

y = 3/4x + b, (x, y)

-5 = 3/4(8) + b

-5 = 6 + b

-5 - 6 = 6 - 6 + b

-5 - 6 = b

-11 = b

y = 3/4x - 11

Answer:

answer is 5

Step-by-step explanation:

I think so

if yes L I k e

Similar triangles are those triangles whose corresponding sides are in the same ratio. The correct option is A.

Similar triangles are those triangles whose corresponding sides are in the same ratio. And the corresponding angles measure the same.

Since for the two of the given triangle, the measure of all the corresponding angles is the same. Also, all the corresponding sides are in a common ratio.

9/3 = 12/4 = 15/5 = 3

Therefore, the two of the given triangles are similar triangles.

Hence, the correct option is A.

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Since AC, MN and DB intersect at O, ∠AOM ≅ NOC and ∠MOD ≅ ∠BON by the vertical angles theorem. It is given that ∠BON ≅ ∠NOC, so by the transitive property of congruence, ∠AOM ≅ ∠MOD

An equation is an expression that shows the relationship between two or more numbers and variables.

An angle is formed from the intersection of two or more lines. Types of angles are acute, obtuse and right.

Since AC, MN and DB intersect at O, ∠AOM ≅ NOC and ∠MOD ≅ ∠BON by the vertical angles theorem. It is given that ∠BON ≅ ∠NOC, so by the transitive property of congruence, ∠AOM ≅ ∠MOD

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Answer:

B. Yes, quadrilateral WXYZ is rhombus because the sides are perpendicular.

Step-by-step explanation:

1) according to the given coordinates the equations of the sides are:

YZ: y=2; WZ: x=2; WX: y=10; XY: x=10. It means

2) YZ⊥WZ; YZ⊥XY; WX⊥XY; WX⊥WZ. To the additional, WX=WZ=YZ=XY, then

3) the given quadrilateral is sqare (rhombus with angles m∠=90°).

4) finally, the correct answer is

B. Yes, quadrilateral WXYZ is rhombus because the sides are perpendicular.

The attached graph represents the graph of the parallel lines

The equation of the lines are linear equation

A linear equation is represented as:

y = mx + b

Where b represents the y-intercept.

Using the above form of linear equation as a guide, the equations of the lines are

y = -x + 5

y = -x + 2

y = -x - 3

y = -x - 2

Where -5, 2, -3 and -2 are the given y-intercepts and y = -x is the base equation.

Next, we plot the equations on a coordinate grid

See attachment

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The height of the pole is 6.50 meters

To calculate the height of the pole, we will use of the following equivalent ratios.

By Pythagoras theorem, we have:

[tex]x ^2 = 2^2 + 3^2\\\\x ^2 = 4 + 9\\\\x = \sqrt{13}[/tex]

So, we have:

[tex]h = \dfrac{\sqrt{13}^2 }{2} \\\\h = \dfrac{13}{2}\\\\h = 6.5[/tex]

Hence, the height of the pole is 6.50 meters

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The correct answer is option D.It is the graph that represents two quantities in a proportional relationship.

A graph is the representation of the data on the vertical and horizontal coordinates so we can see the trend of the data.

As we can see graph D is showing the correct proportionality. We can also create an equation of proportionality as:-

Y = 2x

Put y = 0.5

0.5 = 2 x 0.5 = 1

As we put y=0.5 we get the value of x = 1 as shown in the graph.

Therefore the correct answer is option D.It is the graph that represents two quantities in a proportional relationship.

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Using a system of equations, the total number of marbles that they have is given by:

A. 27.

A system of equations is when two or more variables are related, and equations are built to find the values of each variable.

The variables are given as follows:

Eve has two more marbles than Solene, hence:

x = y + 2.

Solene has twice as many marbles as Steve, hence:

Steve has 7 less marbles than eve, hence:

z = x - 7.

Hence:

[tex]\frac{y}{2} = x - 7[/tex]

y = 2(x - 7)

y = 2(y + 2 - 7)

y = 2y - 10

y = 10.

Then:

Hence the total is given by:

T = 12 + 10 + 5 = 27.

Which means that option A is correct.

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Answer:

Step-by-step explanation:

2 ( x + 3 ) / 4 ( x + 1 )

2 (x + 3) ÷4 (x + 1)

2 (x + 3) ÷ 2×2(x+1)

cancel out the 2 in the numerator with the 2 in the denominator

(x + 3) ÷ 2(x + 1)

x+3 / 2(x+1)