Answer to Question #266090 in Algebra for Israel

Question

1. Given a flow diagram, Calculate the values of the output: 2,5,4,3,9.
2. If A=2 and B=5, Determine the values of a^2+ab+b^2.
3. Expand and calculate: 2^2 x 3^2.
4. Calculate: 5x (-5) + 6x (-8) -16.
5. Simplify: a) 4x (x + 2); b) (3x + 5)(2x + 3); c) 27x y^3 /3x^3 y^2.

6. Solve for x: a) 10x - 6= 5x - 4; b) 4x (2x - 1) = 2(3 - x).

7. Given a triangle KLM with an unknown angle, calculate the value of x if K= 12, L= 23, and M= x.

Answer 1

1. There is no the diagram
2. a^2 + ab + b^2 =$a^2 + ab + b^2 = 2^2 + 5 \cdot 2 + 5^2 = 2\cdot2 + 5\cdot2 + 5\cdot5 = 4 + 10 + 25 = 14 + 25 = 39$
3. $2^2 \cdot 3^2 = (2\cdot2) \cdot (3\cdot3) = 4 \cdot 9 = 36$
4. $5\cdot(-5) + 6\cdot(-8) - 16 = -25 + (-48) - 16 = -25 - 48 - 16 = -73 - 16 = -89$
5. a) if 4x (x + 2) is $4\cdot(x+2)$, then $4\cdot(x+2) = 4\cdot x + 4\cdot2$ (by distributivity property) $=4x + 8$

if 4x (x + 2) is $4x\cdot(x+2)$ , then $4x\cdot(x+2) = 4x\cdot x + 4x\cdot2 = 4x^2 + 8x$ ($4x\cdot2 = 4\cdot2\cdot x$ by associativity property)

b) $(3x + 5)(2x + 3) = 3x\cdot(2x + 3) + 5\cdot(2x + 3) = 3x\cdot2x + 3x\cdot3 + 5\cdot2x + 5\cdot3 = 6x^2 + 9x + 10x + 15 = 6x^2 + 19x + 15$

c) if 27x y^3 /3x^3 y^2 is $\frac{27\cdot y^3} {3x^3 y^2}$ then $\frac{27\cdot y^3} {3x^3 y^2} = /\frac{y^3}{y^2} = y/ = \frac{27\cdot y} {3x^3} = /\frac{27}{3} = 9/ = \frac{9\cdot y} {x^3}$

if 27x y^3 /3x^3 y^2 is $\frac{27x\cdot y^3} {3x^3 y^2}$ then $\frac{27x\cdot y^3} {3x^3 y^2} = \frac{27x\cdot y}{3x^3} = \frac{9x\cdot y}{x^3} = \frac{9y}{x^2}$

6. a)

$$10x - 6= 5x - 4 \\ 10x - 5x = -4 + 6 \\ 5x = 2 \\ x = \frac{2}{5} = 0.4$$

b) if 4x (2x - 1) is $4x (2x - 1)$ then:

$$4x (2x - 1) = 2(3 - x) \\ 4x \cdot 2x - 4x = 2 \cdot 3 - 2\cdot x \\ 8x^2 - 4x = 6 - 2x \\ 8x^2 - 4x + 2x - 6 = 0 \\ 8x^2 - 2x - 6 = 0 |:2 \\ 4x^2 - x - 3 = 0 \\ D = (-1)^2 - 4\cdot4\cdot(-3) = 1 + 48 = 49 = 7^2 \\ x_1 = \frac{-(-1) + 7}{2\cdot4} = \frac{1 + 7}{8} = \frac{8}{8} = 1 \\ x_2 = \frac{-(-1) - 7}{2\cdot4} = \frac{1 - 7}{8} = \frac{-6}{8} = \frac{-3}{4} = -0.75$$

if 4x (2x - 1) is $4\cdot (2x - 1)$ then:

$$4\cdot (2x - 1) = 2(3 - x) \\ 4\cdot2x - 4 \cdot 1 = 2 \cdot 3 - 2 \cdot x \\ 8x - 4 = 6 - 2x \\ 8x + 2x = 6 + 4 \\ 10x = 10 \\ x = 10 : 10 = 1$$

7. Sum of triangle's angles is $180\degree$. Then

$$\angle K + \angle L + \angle M = 180\degree \\ 12\degree + 23\degree + x = 180\degree \\ 35\degree + x = 180\degree \\ x = 180\degree - 35\degree = 145\degree$$

Answers:

1. -
2. 39
3. 36
4. -89
5. a) $4x + 8$ OR a) $4x^2 + 8x$ -- depends on the condition

b) $6x^2 + 19x + 15$

c) $\frac{9\cdot y} {x^3}$ OR c) $\frac{9y}{x^2}$ -- depends on the condition

6. a) x = 2/5 = 0,4

b) x₁ = 1, x₂ = -0,75 OR b) x = 1 -- depends on the condition

7. 145