$\bf \textit{vertex of a vertical parabola, using coefficients} \\\\ f(x)=\stackrel{\stackrel{a}{\downarrow }}{2}x^2\stackrel{\stackrel{b}{\downarrow }}{-8}x\stackrel{\stackrel{c}{\downarrow }}{+9} \qquad \qquad \left(-\cfrac{ b}{2 a}~~~~ ,~~~~ c-\cfrac{ b^2}{4 a}\right) \\\\\\ \left(-\cfrac{-8}{2(2)}~~,~~9-\cfrac{(-8)^2}{4(2)} \right)\implies \left( \cfrac{8}{4}~~,~~9-\cfrac{64}{8} \right) \\\\\\ (2~~,~~9-8)\implies (2~~,~~1)$

5 0

4 years ago

a) If the price of a train ticket from New York to Washington DC rose by the rate of inflation between 1995 and 2016, how much w

Alla [95]

Answer:

$ 99.60

Step-by-step explanation:

Thinking process:

According to financial data, the inflation rate in 1995 was at 2.5 %.

In 2016, the inflation rate was at 2.1 %

The difference was: 2.1 - 2.5 = -0.4 %

The price in 2016 = Original price - Adjusted price

Adjusted price = $0.004* ($ 100) = $ 0.4$

The new price will be $$ 100 - $ 0.4\\= $ 99.60$

The new price will be $ 99.60

4 0

4 years ago

Simplify (x + 2/ x^2 + 2x -3) / (x + 2/x^2 - x)

rjkz [21]

Answer:

The simplest form is x/(x + 3)

Step-by-step explanation:

* To simplify the rational Expression lets revise the factorization

of the quadratic expression

* To factor a quadratic in the form x² ± bx ± c:

- First look at the c term

# If the c term is a positive number, and its factors are r and s they

will have the same sign and their sum is b.

# If the c term is a negative number, then either r or s will be negative

but not both and their difference is b.

- Second look at the b term.

# If the c term is positive and the b term is positive, then both r and

s are positive.

Ex: x² + 5x + 6 = (x + 3)(x + 2)

# If the c term is positive and the b term is negative, then both r and s

are negative.

Ex: x² - 5x + 6 = (x -3)(x - 2)

# If the c term is negative and the b term is positive, then the factor

that is positive will have the greater absolute value. That is, if

|r| > |s|, then r is positive and s is negative.

Ex: x² + 5x - 6 = (x + 6)(x - 1)

# If the c term is negative and the b term is negative, then the factor

that is negative will have the greater absolute value. That is, if

|r| > |s|, then r is negative and s is positive.

Ex: x² - 5x - 6 = (x - 6)(x + 1)

* Now lets solve the problem

- We have two fractions over each other

- Lets simplify the numerator

∵ The numerator is $\frac{x+2}{x^{2}+2x-3}$

- Factorize its denominator

∵ The denominator = x² + 2x - 3

- The last term is negative then the two brackets have different signs

∵ 3 = 3 × 1

∵ 3 - 1 = 2

∵ The middle term is +ve

∴ -3 = 3 × -1 ⇒ the greatest is +ve

∴ x² + 2x - 3 = (x + 3)(x - 1)

∴ The numerator = $\frac{(x+2)}{(x+3)(x-2)}$

- Lets simplify the denominator

∵ The denominator is $\frac{x+2}{x^{2}-x}$

- Factorize its denominator

∵ The denominator = x² - 2x

- Take x as a common factor and divide each term by x

∵ x² ÷ x = x

∵ -x ÷ x = -1

∴ x² - 2x = x(x - 1)

∴ The denominator = $\frac{(x+2)}{x(x-1)}$

* Now lets write the fraction as a division

∴ The fraction = $\frac{x+2}{(x+3)(x-1)}$ ÷ $\frac{x+2}{x(x-1)}$

- Change the sign of division and reverse the fraction after it

∴ The fraction = $\frac{(x+2)}{(x+3)(x-1)}*\frac{x(x-1)}{(x+2)}$

* Now we can cancel the bracket (x + 2) up with same bracket down

and cancel bracket (x - 1) up with same bracket down

∴ The simplest form = $\frac{x}{x+3}$

5 0

4 years ago

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