HiSET - Algebraic Concepts

What is the vertex of the following quadratic polynomial?

$(\frac{3}{4},\frac{127}{8})$

$(-\frac{5}{3},\frac{12}{7})$

$(\frac{4}{3},\frac{98}{17})$

$(-\frac{4}{9},-\frac{58}{17})$

$(-\frac{4}{23},-\frac{15}{17})$

Explanation

Given a quadratic function

the vertex will always be

$(-\frac{b}{2a},\frac{4ac-b^2}{4a})$ .

Thus, since our function is

, , and .

We plug these variables into the formula to get the vertex as

$(-\frac{-3}{2\times2},\frac{4\times2\times17-(-3)^2}{4\times2})$

$(-\frac{-3}{4},\frac{136-9}{8})$

$=(\frac{3}{4},\frac{127}{8})$ .

Hence, the vertex of

$(\frac{3}{4},\frac{127}{8})$ .

Solve for :

$\frac{y+1}{x+1} = 3$

$x = \frac{1}{3} y - \frac{2}{3}$

$x = \frac{1}{3} y + \frac{4}{3}$

$x = \frac{2}{3} y -\frac{4}{3}$

$x = \frac{2}{3} y + \frac{1}{3}$

$x = \frac{4}{3} y + \frac{1}{3}$

Explanation

To solve for in a literal equation, use the properties of algebra to isolate on one side, just as if you were solving a regular equation.

First, take the reciprocal of both sides:

$\frac{y+1}{x+1} = 3$

$\frac{x+1}{y+1} = \frac{1}{3}$

Multiply both sides by :

$\frac{x+1}{y+1} \cdot (y+1)= \frac{1}{3}(y+1)$

$x+1= \frac{1}{3}(y+1)$

Distribute on the right:

$x+1= \frac{1}{3} \cdot y+ \frac{1}{3} \cdot 1$

$x+1= \frac{1}{3} y+ \frac{1}{3}$

Subtract 1 from both sides, rewriting 1 as $\frac{3}{3}$ to facilitate subtraction:

$x+1 - 1 = \frac{1}{3} y+ \frac{1}{3} - 1$

$x = \frac{1}{3} y+ \frac{1}{3} - \frac{3}{3}$

$x = \frac{1}{3} y - \frac{2}{3}$ ,

the correct response.

The graph of a function is shown below, with labels on the y-axis hidden.

Graph zeroes 2

Determine which of the following functions best fits the graph above.

$f(x)=\frac{x+1}{x+2}$

$f(x)=x^2x^{-2}$

Explanation

Use the zeroes of the graph to determine the matching function. Zeroes are values of x where . In other words, they are points on the graph where the curve touches zero.

Visually, you can see that the curve crosses the x-axis when , , and . Therefore, you need to look for a function that will equal zero at these x values.

A function with a factor of will equal zero when , because the factor of will equal zero. The matching factors for the other two zeroes, and , are and , respectively.

The answer choice has all of these factors, but it is not the answer because it has an additional zero that would be visible on the graph. Notice it has a factor of , which results in a zero at . This additional zero that isn't present in the graph indicates that this cannot be matching function.

is the answer because it has all of the required factors and, as a result, the required zeroes, while not having additional zeroes. Notice that the constant coefficient of negative 2 does not affect where the zeroes are.

What is 25% of ?

Explanation

Solve for in the equation

by isolating on the left side. Do this by reversing the operations in the reverse of the order of operations.

First, subtract 17 from both sides:

Now, divide both sides by 2:

$2x \div 2 = 82 \div 2$

One way to find 25% of this value is to multiply 41 by 25 and divide by 100:

$\frac{41 \times 25}{100} = \frac{1,025}{100} = 10.25$ ,

the correct choice.

Solve for :

$\frac{y+1}{x+1} = 3$

$x = \frac{1}{3} y - \frac{2}{3}$

$x = \frac{1}{3} y + \frac{4}{3}$

$x = \frac{2}{3} y -\frac{4}{3}$

$x = \frac{2}{3} y + \frac{1}{3}$

$x = \frac{4}{3} y + \frac{1}{3}$

Explanation

To solve for in a literal equation, use the properties of algebra to isolate on one side, just as if you were solving a regular equation.

First, take the reciprocal of both sides:

$\frac{y+1}{x+1} = 3$

$\frac{x+1}{y+1} = \frac{1}{3}$

Multiply both sides by :

$\frac{x+1}{y+1} \cdot (y+1)= \frac{1}{3}(y+1)$

$x+1= \frac{1}{3}(y+1)$

Distribute on the right:

$x+1= \frac{1}{3} \cdot y+ \frac{1}{3} \cdot 1$

$x+1= \frac{1}{3} y+ \frac{1}{3}$

Subtract 1 from both sides, rewriting 1 as $\frac{3}{3}$ to facilitate subtraction:

$x+1 - 1 = \frac{1}{3} y+ \frac{1}{3} - 1$

$x = \frac{1}{3} y+ \frac{1}{3} - \frac{3}{3}$

$x = \frac{1}{3} y - \frac{2}{3}$ ,

the correct response.

Subtract and .

Explanation

Step 1: Subtract these terms by separating by exponents...

Step 2: Add all the simplified terms together...

Solve the following equation:

$\sqrt[3]{5-2x}+5=4$

$x=\sqrt[3]{5}$

Explanation

$\sqrt[3]{5-2x}+5=4$

The first step to solving an equation where is in a radical is to isolate the radical. To do this, we need to subtract the 5 from both sides.

$\sqrt[3]{5-2x}+5-5=4-5$

$\sqrt[3]{5-2x}=-1$

Now that the radical is isolated, clear the radical by raising both sides to the power of 3. Note:

$(-1)^3 = -1 \cdot -1 \cdot -1 = -1$

$(\sqrt[3]{5-2x})^3=(-1)^3$

Now we want to isolate the term. First, subtract the 5 from both sides.

Finally, divide both sides by to solve for .

$\frac{-2x}{-2}=\frac{-6}{-2}$

Add or subtract: $\dfrac {7}{12}-\dfrac {2}{3}+\dfrac {2}{7}$

$\dfrac {17}{84}$

$\dfrac {13}{84}$

$\dfrac {23}{84}$

$\dfrac {4}{21}$

Explanation

Step 1: Find the Least Common Denominator of these fraction. We will list out multiples of each denominator until we find a common number for all three fractions...

$12,24,36,48,60,72,{\color{Red} 84},...$

$7,14,21,28,35,42,...,63,70,77,{\color{Red} 84}$

$3,6,9,...,72,75,78,81,{\color{Red} 84}$

The smallest common denominator is .

Step 2: Since the denominator is , we will convert all denominators to .

$\dfrac {7}{12}=\dfrac {x}{84}\rightarrow \dfrac {7\times 7}{12 \times 7}=\dfrac {x}{84}\rightarrow \dfrac {49}{84}=\dfrac {x}{84}\implies x=49$

$-\dfrac {2}{3}=\dfrac {x}{84}\rightarrow -\dfrac{2\times 28}{3\times 28}=\dfrac {x}{84}\rightarrow -\dfrac {56}{84}=\dfrac {x}{84}\implies x=-56$

$\dfrac {2}{7}=\dfrac {x}{84}\rightarrow \dfrac {2\times12}{7\times 12}=\dfrac {x}{84}\rightarrow \dfrac {24}{84}=\dfrac {x}{84}\implies x=24$

Step 3: Add up all the values of x...

Step 4: The result from step is the numerator and is the denominator. We will put these together.

Final Answer: $\dfrac {17}{84}$

What is the coefficient of the second highest term in the expression: ?

Explanation

Step 1: Rearrange the terms from highest power to lowest power.

We will get: .

Step 2: We count the second term from starting from the left since it is the second highest term in the rearranged expression.

Step 3: Isolate the term.

The second term is

Step 4: Find the coefficient. The coefficient of a term is considered as the number before any variables. In this case, the coefficient is .

So, the answer is .

Identify the coefficients in the following formula:

$\frac{5x^2+2x-4y^3+3y-6}{7\times12+9-13}=29$

All of these

Explanation

Generally speaking, in an equation a coefficient is a constant by which a variable is multiplied. For example, and are coefficients in the following equation: