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Naya [18.7K]
3 years ago
15

Prove that

TexFormula1" title="2 \tan30 \div 1 + tan ^{2} 30 = sin60" alt="2 \tan30 \div 1 + tan ^{2} 30 = sin60" align="absmiddle" class="latex-formula">
prove that

.

​
Mathematics
1 answer:
iVinArrow [24]3 years ago
3 0

Step-by-step explanation:

2tan 30° / 1 + tan² 30° =

2(⅓√3) /1 + (⅓√3)² =

⅔√3 / 1+ ⅓ =

⅔√3 / 4/3 =

2/4 √3 =

½√3 = sin 60° (proven)

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Solve the following system of equations for a and for b:
viktelen [127]

Answer:

a=3\\b=1

Step-by-step explanation:

9a+3b=30\\8a+4b=28

Let's solve the second equation for a to later on replace it in the first equation.

8a+4b=28\\8a=28-4b\\a=\frac{28-4b}{8}

Now plug this into the first equation.

9a+3b=30\\9(\frac{28-4b}{8})+3b=30

Distribute the 9

(\frac{252-36b}{8}) +3b=30

Break down the fraction.

\frac{252}{8}-\frac{36b}{8}+3b=30

Simplify.

\frac{63}{2}-\frac{9}{2}b+3b=30

Subtract \frac{63}{2}

-\frac{9}{2}b+3b=30-\frac{63}{2}

Combine like terms.

\frac{-9+2*3}{2}b=\frac{30*2-63}{2}

\frac{-9+6}{2}b=\frac{60-63}{2}

\frac{-3}{2}b=\frac{-3}{2}

Muliply by the reciprocal or inverted fraction next to b.

(-\frac{2}{3})(-\frac{3}{2}) b=-\frac{3}{2}(-\frac{2}{3})

b=1

Now plug this value into any of the equations to find the value of a.

8a+4b=28\\8a+4(1)=28\\8a+4=28\\8a=28-4\\8a=24\\a=\frac{24}{8}\\ a=3

5 0
4 years ago
The binomial (2x+9) is a factor of a quadratic expression 14x^2+57x-27 what is the other factor of the expression
Blizzard [7]

Answer:

Input:

14 x^2 + 57 x - 27

Plots:

Geometric figure:

parabola

Alternate forms:

(7 x - 3) (2 x + 9)

x (14 x + 57) - 27

14 (x + 57/28)^2 - 4761/56

Roots:

x = -9/2

x = 3/7

Polynomial discriminant:

Δ = 4761

Properties as a real function:

Domain

R (all real numbers)

Range

{y element R : y>=-4761/56}

Derivative:

d/dx(14 x^2 + 57 x - 27) = 28 x + 57

Indefinite integral:

integral(-27 + 57 x + 14 x^2) dx = (14 x^3)/3 + (57 x^2)/2 - 27 x + constant

Global minimum:

min{14 x^2 + 57 x - 27} = -4761/56 at x = -57/28

Definite integral:

integral_(-9/2)^(3/7) (-27 + 57 x + 14 x^2) dx = -109503/392≈-279.344

Definite integral area below the axis between the smallest and largest real roots:

integral_(-9/2)^(3/7) (-27 + 57 x + 14 x^2) θ(27 - 57 x - 14 x^2) dx = -109503/392≈-279.344

Step-by-step explanation:

6 0
4 years ago
Write the polynomial as a square of a binomial or as an expression opposite to a square of a binomial:
spayn [35]

Answer:

A) 0.25x^2-0.6xy+0.36y^2=\left(0.5x-0.6y\right)^2

B) -a^2+0.6a-0.09=-\left(10a-3\right)^2

C) \frac{9a^4}{16}+a^3+\frac{4a^2}{9}=a^2(\left(9a+8\right)^2)

D) -16m^2-24mn -9n^2=-\left(4m+3n\right)^2

Step-by-step explanation:

The square of a binomial is the sum of: the square of the first terms, twice the product of the two terms, and the square of the last term.

(a+b)^2 = a^2 + 2ab + b^2\\\\(a-b)^2 = a^2 - 2ab + b^2

To find the square of the binomial of the following polynomials you must:

A) 0.25x^2-0.6xy+0.36y^2

Apply radical rule: a=\left(\sqrt{a}\right)^2

0.25=\left(\sqrt{0.25}\right)^2\\0.36=\left(\sqrt{0.36}\right)^2

\left(\sqrt{0.25}\right)^2x^2-0.6xy+\left(\sqrt{0.36}\right)^2y^2

Apply exponent rule: a^mb^m=\left(ab\right)^m

\left(\sqrt{0.25}\right)^2x^2=\left(\sqrt{0.25}x\right)^2\\\left(\sqrt{0.36}\right)^2y^2=\left(\sqrt{0.36}y\right)^2

\left(\sqrt{0.25}x\right)^2-0.6xy+\left(\sqrt{0.36}y\right)^2

Rewrite 0.6xy as 2\cdot \:0.5x\cdot \:0.6y

\left(\sqrt{0.25}x\right)^2-2\cdot \:0.5x\cdot \:0.6y+\left(\sqrt{0.36}y\right)^2

Apply perfect square formula: \left(a-b\right)^2=a^2-2ab+b^2

a=0.5x,\:b=0.6y

\left(\sqrt{0.25}x\right)^2-2\cdot \:0.5x\cdot \:0.6y+\left(\sqrt{0.36}y\right)^2=\left(0.5x-0.6y\right)^2

B) -a^2+0.6a-0.09

Multiply both sides by 100

-a^2\cdot \:100+0.6a\cdot \:100-0.09\cdot \:100\\-100a^2+60a-9

Factor out common term -1

-\left(100a^2-60a+9\right)

Break the expression into groups and factor out common terms

-(\left(100a^2-30a\right)+\left(-30a+9\right))\\-(10a\left(10a-3\right)-3\left(10a-3\right))\\-(\left(10a-3\right)\left(10a-3\right))\\-\left(10a-3\right)^2

C) \frac{9a^4}{16}+a^3+\frac{4a^2}{9}

Apply exponent rule: a^{b+c}=a^ba^c

a^3=aa^2\\a^4=a^2a^2

\frac{9a^2a^2}{16}+aa^2+\frac{4a^2}{9}

Factor out common term a^2

a^2\left(\frac{9a^2}{16}+a+\frac{4}{9}\right)

Factor \frac{9a^2}{16}+a+\frac{4}{9}\right

Find the Least Common Multiplier (LCM) of 16, 9 which is 144.

Multiply by LCM

\frac{9a^2}{16}\cdot \:144+a\cdot \:144+\frac{4}{9}\cdot \:144\\81a^2+144a+64

81a^2+144a+64=\left(9a\right)^2+2\cdot \:9a\cdot \:8+8^2

Apply perfect square formula: \left(a+b\right)^2=a^2+2ab+b^2

a=9a,\:b=8

81a^2+144a+64=\left(9a+8\right)^2

\frac{9a^4}{16}+a^3+\frac{4a^2}{9}=a^2(\left(9a+8\right)^2)

D) -16m^2-24mn -9n^2

Factor out common term -1

-\left(16m^2+24mn+9n^2\right)

Break the expression into groups and factor out common terms

\left(16m^2+12mn\right)+\left(12mn+9n^2\right)\\4m\left(4m+3n\right)+3n\left(4m+3n\right)\\\left(4m+3n\right)\left(4m+3n\right)\\-\left(4m+3n\right)\left(4m+3n\right)\\-\left(4m+3n\right)^2

-16m^2-24mn -9n^2=-\left(4m+3n\right)^2

3 0
4 years ago
Please help me with number 12 I’m begging you please help me please I’m struggling so much with this question please help me I’m
ASHA 777 [7]

Answer:

6/4 miles or write like 1 1/2miles.

Step-by-step explanation:

Half way to his friend's home is 3/4 miles.

So the total miles should be:

3/4+3/4=6/4=1 1/2

5 0
3 years ago
Find the value of x and y<br><br> 2x+7y=4<br> -4x-3y=14
guapka [62]

Answer:

x=-5 y=2

Step-by-step explanation:

1) multiplying equation 1 by 4 and multiplying equation 2 by 2 .

2)adding them we get value of y.

3) replacing value of y in either of given equation

5 0
3 years ago
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