Question

What is the range of \(\\{-3,0,2, x, 6, y, 3\\}\) if \(x^{2}+8 x=-16\) and \(y^{2}-y=\) \(6 ?\)

Short answer

The range is 10.

Step-by-step solution

- Solve for x

To find the value of x, use the given equation: \[ x^{2} + 8 x = -16 \]Rearrange to form a quadratic equation:\[ x^{2} + 8 x + 16 = 0 \]Factorize the quadratic equation:\[ (x + 4)^{2} = 0 \]So, \( x = -4 \)

- Solve for y

To find the value of y, use the given equation:\[ y^{2} - y = 6 \]Rearrange to form a quadratic equation:\[ y^{2} - y - 6 = 0 \]Factorize the quadratic equation:\[ (y - 3)(y + 2) = 0 \]This gives two solutions: \[ y = 3 \] and \[ y = -2 \]

- List the Values

Substitute the values of x and y back into the set:\[ \{-3, 0, 2, -4, 6, 3, 3\} \] and \[ \{-3, 0, 2, -4, 6, -2, 3\} \]

- Find the Range

To find the range, identify the smallest and largest numbers in each set. The minimum value is -4 in both sets, the maximum values are 6 and 6 respectively. So, the range for both sets is: \[ 6 - (-4) = 10 \]

Key concepts

Quadratic Equations

Quadratic equations are polynomial equations of the second degree. They have the form \(ax^2 + bx + c = 0\), where \(x\) represents an unknown variable, and \(a\), \(b\), and \(c\) are known coefficients with \(a e 0\). Solving quadratic equations often involves re-arranging the equation to standard form and applying techniques like factoring, completing the square, or using the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]. Understanding these methods is crucial for solving many GRE mathematics problems.

Algebraic Solutions

Algebra involves the manipulation of mathematical symbols to solve equations. For quadratic equations, algebraic solutions include rearranging the equation to its standard form and using factoring techniques. For example, the equation \(x^2 + 8x = -16\) is transformed into \(x^2 + 8x + 16 = 0\). This can then be factored into \((x + 4)^2 = 0\), giving us the solution \(x = -4\). Similarly, \(y^2 - y - 6 = 0\) can be factored into \((y - 3)(y + 2) = 0\), providing the solutions \(y = 3\) and \(y = -2\).

Range Calculation

The range of a set of numbers is the difference between the largest and smallest values in that set. To calculate the range, you need to identify these two extremes. For example, in the set \{-3, 0, 2, -4, 6, 3, 3\}, the minimum value is \(-4\) and the maximum value is \(6\). Therefore, the range is calculated as:

\[6 - (-4) = 6 + 4 = 10\].
Range is an important concept in GRE prep as it helps in understanding the spread and distribution of data.

GRE Prep

Preparing for the GRE involves mastering various mathematical concepts and problem-solving techniques. Focusing on core areas like quadratic equations, algebraic solutions, and range calculation is essential. Practice solving different types of equations, familiarize yourself with common mathematical operations, and work on timed practice problems to improve speed and accuracy. Using efficient problem-solving strategies can help you perform better in the quantitative section of the GRE.