Problem 5
Question
Solve the quadratic inequality. $$3 x^{2} \geq 2 x+5$$
Step-by-Step Solution
Verified Answer
The solution for the quadratic inequality \(3x^2 \geq 2x + 5\) is \(x \in (-\infty, -1] \cup [1.67, \infty)\)
1Step 1: Rearrange the inequality
First, rearrange the inequality into standard quadratic form by subtracting \(2x + 5\) from both sides to get \(3x^2 - 2x - 5 \geq 0\)
2Step 2: Factorize the quadratic
Next, it's necessary to factor the quadratic, so it's easier to find the roots. However, this is a quadratic with integer coefficients, but non-integer roots. Therefore, it's best to use the quadratic formula to find the roots. Because the general formula for a quadratic equation is \(ax^2 + bx + c = 0\), in our case, \(a = 3\), \(b = -2\), \(c = -5\). Inserting these numbers into the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) gives \(x = \frac{2 \pm \sqrt{(-2)^2 - 4*3*(-5)}}{2*3} = \frac{2 \pm \sqrt{4 + 60}}{6} = \frac{2 \pm \sqrt{64}}{6} = \frac{2 \pm 8}{6}\). So the solutions are \(x = 1.67\) and \(x = -1\).
3Step 3: Find the solution set
Now that the roots of the quadratic are known, it is possible to find the solution set of the inequality. To do that one has to test the signs of each of three intervals determined by the roots. To test an interval, one can pick any number within the interval and substitute it into the quadratic form of the inequality. If the output is greater than or equal to zero, then that interval is in the solution set. The intervals are: \(-\infty, -1\), \(-1, 1.67\), and \(1.67, \infty\). When tested these intervals show that the output is positive for the intervals \(-\infty, -1\) and \(1.67, \infty\). Therefore, the solution set of the inequality is \(x \in (-\infty, -1] \cup [1.67, \infty)\)
Key Concepts
Quadratic EquationFactoring QuadraticsQuadratic FormulaInequality SolvingSolution Set Analysis
Quadratic Equation
A quadratic equation is a polynomial equation of the second degree. It generally takes the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents an unknown. Quadratics have distinctive properties including a symmetrical U-shaped graph called a parabola.
They are foundational in different areas of math and can have up to two roots or solutions, which are crucial to solving quadratic inequalities.
To tackle a quadratic inequality, like \( 3x^2 \geq 2x + 5 \), we first rearrange it into standard quadratic form: \( 3x^2 - 2x - 5 \geq 0 \). This step is the foundation for further steps such as factoring or applying the quadratic formula.
They are foundational in different areas of math and can have up to two roots or solutions, which are crucial to solving quadratic inequalities.
To tackle a quadratic inequality, like \( 3x^2 \geq 2x + 5 \), we first rearrange it into standard quadratic form: \( 3x^2 - 2x - 5 \geq 0 \). This step is the foundation for further steps such as factoring or applying the quadratic formula.
Factoring Quadratics
Factoring quadratics involves rewriting the quadratic expression as a product of two binomial expressions. This approach provides insights into the roots of the quadratic equation. However, not all quadratics can be easily factored, especially when they have non-integer roots.
In the given problem, it is not straightforward to factor \( 3x^2 - 2x - 5 \) due to its complex roots. When factoring is difficult, alternative methods such as the quadratic formula become necessary.
Factoring is useful because it simplifies the expression and helps identify intervals to test in the inequality.
In the given problem, it is not straightforward to factor \( 3x^2 - 2x - 5 \) due to its complex roots. When factoring is difficult, alternative methods such as the quadratic formula become necessary.
Factoring is useful because it simplifies the expression and helps identify intervals to test in the inequality.
Quadratic Formula
The quadratic formula is a reliable method for finding the roots of any quadratic equation. It is expressed as:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
This formula helps solve quadratics that are not easily factored, like in our example where \( a = 3 \), \( b = -2 \), and \( c = -5 \).
By substituting these values into the formula, we calculate the roots as \( x = 1.67 \) and \( x = -1 \).
These roots are crucial, as they define the critical points that divide the number line into test intervals for solving the inequality.
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
This formula helps solve quadratics that are not easily factored, like in our example where \( a = 3 \), \( b = -2 \), and \( c = -5 \).
By substituting these values into the formula, we calculate the roots as \( x = 1.67 \) and \( x = -1 \).
These roots are crucial, as they define the critical points that divide the number line into test intervals for solving the inequality.
Inequality Solving
Solving quadratic inequalities extends the process of finding roots by determining where the quadratic expression is positive or negative.
With roots like \( x = -1 \) and \( x = 1.67 \), they partition the number line into intervals:
Such interval testing uncovers the range of solutions satisfying the original quadratic inequality.
With roots like \( x = -1 \) and \( x = 1.67 \), they partition the number line into intervals:
- \((-\infty, -1)\)
- \((-1, 1.67)\)
- \((1.67, \infty)\)
Such interval testing uncovers the range of solutions satisfying the original quadratic inequality.
Solution Set Analysis
Analyzing the solution set involves collating the intervals where the quadratic expression meets the inequality requirement.
From our investigation, the valid intervals are \((-\infty, -1]\) and \([1.67, \infty)\), indicating where the expression is non-negative.
Thus, the solution set for the inequality is \(x \in (-\infty, -1] \cup [1.67, \infty)\).
This final step underscores the importance of correctly identifying and testing interval sections to ensure each resolves the inequality.
It helps visualize and comprehend how the roots influence the behavior of the quadratic across its domain, informing the correct solution set.
From our investigation, the valid intervals are \((-\infty, -1]\) and \([1.67, \infty)\), indicating where the expression is non-negative.
Thus, the solution set for the inequality is \(x \in (-\infty, -1] \cup [1.67, \infty)\).
This final step underscores the importance of correctly identifying and testing interval sections to ensure each resolves the inequality.
It helps visualize and comprehend how the roots influence the behavior of the quadratic across its domain, informing the correct solution set.
Other exercises in this chapter
Problem 4
Classfy each function as odd, even, or neither. $$h(x)=3|x|$$
View solution Problem 4
Complete them to review topics relevant to the remaining exercises. Find the \(x\) -intercept of \(f(x)=3 x+9\).
View solution Problem 5
These exercises correspond to the Just in Time references in this section. Complete them to review topics relevant to the remaining exercises. Find the conjugat
View solution Problem 5
For each polynomial, determine which of the numbers listed next to it are zeros of the polynomial. $$p(x)=(x-10)^{8}, x=6,-10,10$$
View solution