Solve the Inequality for z z^2-12z+36> =0
The given problem requests to find the set of all values of variable 'z' that satisfy the inequality z^2 - 12z + 36 ≥ 0. The inequality involves a quadratic expression, and solving it entails determining for which values of 'z' the quadratic is greater than or equal to zero. This typically involves factoring the quadratic (if possible), finding its critical points (the values of 'z' where the expression equals zero), and performing a sign analysis to understand where the quadratic expression is positive (above the x-axis) or non-negative, thereby determining the regions where the inequality holds true.
$z^{2} - 12 z + 36 \geq 0$
Answer
Solution:
Step 1:
Transform the inequality into an equivalent equation: $z^2 - 12z + 36 = 0$
Step 2:
Apply the perfect square factorization method.
Step 2.1:
Express $36$ as the square of $6$: $z^2 - 12z + 6^2 = 0$
Step 2.2:
Confirm that the middle term equals twice the product of the square roots of the first and last terms: $12z = 2 \cdot z \cdot 6$
Step 2.3:
Reformulate the quadratic expression: $z^2 - 2 \cdot z \cdot 6 + 6^2 = 0$
Step 2.4:
Factor by applying the formula for a perfect square trinomial $a^2 - 2ab + b^2 = (a - b)^2$, where $a = z$ and $b = 6$: $(z - 6)^2 = 0$
Step 3:
Isolate $z$ by setting the expression inside the square equal to zero: $z - 6 = 0$
Step 4:
Solve for $z$ by adding $6$ to both sides: $z = 6$
Step 5:
Formulate test intervals around the root: $z < 6$ and $z > 6$
Step 6:
Select test points from each interval to verify the inequality.
Step 6.1:
For the interval $z < 6$, choose a test value to check the inequality.
Step 6.1.1:
Pick a test point within the interval $z < 6$: $z = 0$
Step 6.1.2:
Substitute $z = 0$ into the original inequality: $0^2 - 12 \cdot 0 + 36 \geq 0$
Step 6.1.3:
Evaluate the inequality: $36 \geq 0$ is true, indicating the interval is valid.
Step 6.2:
For the interval $z > 6$, select a test value to check the inequality.
Step 6.2.1:
Choose a test point within the interval $z > 6$: $z = 8$
Step 6.2.2:
Substitute $z = 8$ into the original inequality: $8^2 - 12 \cdot 8 + 36 \geq 0$
Step 6.2.3:
Evaluate the inequality: $4 \geq 0$ is true, confirming the interval is valid.
Step 6.3:
Assess the intervals to determine which satisfy the inequality: both $z < 6$ and $z > 6$ are true.
Step 7:
The solution is the union of all true intervals: $z \leq 6$ or $z \geq 6$
Step 8:
Merge the intervals to express the solution: all real numbers are solutions.
Step 9:
The final solution can be represented in various ways: all real numbers or using interval notation $(-\infty, \infty)$
Knowledge Notes:
The problem involves solving a quadratic inequality. The steps taken to solve such inequalities typically include:
Converting the inequality into an equation to find critical values (roots).
Factoring the quadratic expression, if possible, to simplify the process.
Using the perfect square trinomial rule, which states that $a^2 - 2ab + b^2 = (a - b)^2$, to factor the quadratic expression.
Determining the intervals around the roots to test the inequality.
Selecting test values from each interval to check if they satisfy the original inequality.
Combining the intervals that satisfy the inequality to form the solution set.
Representing the solution set in various formats, such as a written description, set builder notation, or interval notation.
In this particular problem, the quadratic expression is a perfect square, meaning it factors into the square of a binomial. The inequality is satisfied for all real numbers since the expression $(z - 6)^2$ is always non-negative.
