Master 12.7.9 Practice Quiz Today

Subtracting 5 from both sides gives 2x = 8, and dividing by 2 yields x = 4. This straightforward process illustrates the basic method of solving a linear equation.

The slope is calculated by taking the difference in the y-coordinates divided by the difference in the x-coordinates. Here, (11 - 3)/(5 - 2) simplifies to 8/3, which is the correct value.

A quadratic function is typically written in the form ax^2 + bx + c, where a is non-zero. The equation y = x^2 + 3x + 2 fits this format, making it the correct answer.

In Euclidean geometry, the interior angles of any triangle always add up to 180 degrees. This is a fundamental property used in many geometric proofs and problems.

Taking the square root of both sides of the equation x² = 16 results in x = ±4. Both -4 and 4 are valid solutions within the real number system.

By rearranging the second equation, we have y = x - 1. Substituting this into the first equation yields 2x + (x - 1) = 7, simplifying to 3x = 8 and therefore x = 8/3.

Substituting x = 4 into the function gives f(4) = 3(4) - 5, which simplifies to 12 - 5 = 7. This demonstrates the evaluation of a linear function at a specific point.

The vertex form y = a(x - h)^2 + k directly shows the vertex of the parabola as (h, k). This form is particularly useful for graphing and analyzing quadratic functions.

Factoring the quadratic as (x - 2)(x - 3) = 0 produces the solutions x = 2 and x = 3. This method of factoring is a standard approach for solving quadratic equations.

First, compute f(2) which is 2² = 4. Then, f(f(2)) equals f(4) which is 4² = 16, demonstrating the concept of function composition.

The circumference of a circle is calculated using the formula 2πr. With r = 5, the circumference is 2π � - 5 = 10π.

Since the ratio of the sides is 1:2, every side in the larger triangle is twice as long as the corresponding side in the smaller triangle. Multiplying 3 by 2 gives 6 as the correct length.

A trigonometric identity is an equality that holds true for all values of the variable within its domain. The equation sin²θ + cos²θ = 1 is one of the most fundamental identities in trigonometry.

Since 2 raised to the power of 3 equals 8, log₂(8) evaluates to 3. This is a basic logarithmic calculation that reinforces the definition of a logarithm.

The absolute value equation |x - 3| = 5 splits into two cases: x - 3 = 5 and x - 3 = -5. Solving these gives x = 8 and x = -2 respectively.

Setting the equations equal gives x² = 4x - x², which simplifies to 2x² - 4x = 0. Factoring out 2x results in 2x(x - 2) = 0, so the solutions are x = 0 and x = 2.

Using the logarithmic property, the equation becomes log₃(x(x - 2)) = 1, implying that x(x - 2) = 3. Solving the quadratic x² - 2x - 3 = 0 yields x = 3 (after dismissing the extraneous solution based on the domain).

A discriminant of zero in a quadratic function indicates that there is exactly one unique real solution or a repeated root. This means the parabola touches the x-axis at its vertex but does not cross it.

Expanding (√2 + √3)² using the formula (a + b)² = a² + 2ab + b² results in 2 + 2√6 + 3. Combining like terms yields 5 + 2√6 as the fully expanded form.

By rewriting the function as √2 sin(x - π/4), the maximum value of the sine function, which is 1, gives a maximum of √2. This transformation simplifies finding the maximum of the original function.