$4 x^2-8 x+3 \leqslant 0$
Answer (1)
Factor the quadratic expression: 4 x 2 − 8 x + 3 = ( 2 x − 1 ) ( 2 x − 3 ) .
Find the roots of the equation ( 2 x − 1 ) ( 2 x − 3 ) = 0 , which are x = 2 1 and x = 2 3 .
Determine the interval where the quadratic is less than or equal to zero. Since the parabola opens upwards, the solution is between the roots.
The solution to the inequality is [ 2 1 , 2 3 ] .
Explanation
Problem Analysis We are given the quadratic inequality 4 x 2 − 8 x + 3 ≤ 0 . Our goal is to find the interval of x values that satisfy this inequality.
Factoring the Quadratic First, let's factor the quadratic expression. We are looking for two numbers that multiply to 4 × 3 = 12 and add up to − 8 . These numbers are − 2 and − 6 . So we can rewrite the middle term as − 2 x − 6 x :
4 x 2 − 6 x − 2 x + 3 ≤ 0 Now, we factor by grouping: 2 x ( 2 x − 3 ) − 1 ( 2 x − 3 ) ≤ 0 ( 2 x − 1 ) ( 2 x − 3 ) ≤ 0
Finding the Roots Next, we find the roots of the quadratic equation ( 2 x − 1 ) ( 2 x − 3 ) = 0 . The roots are the values of x that make each factor equal to zero: 2 x − 1 = 0 ⇒ x = 2 1 2 x − 3 = 0 ⇒ x = 2 3
Determining the Interval Now, we determine the interval where the quadratic expression is less than or equal to zero. Since the coefficient of the x 2 term is positive (4), the parabola opens upwards. This means the quadratic expression is negative between the roots. Therefore, the solution to the inequality is the interval between the roots, including the roots themselves: 2 1 ≤ x ≤ 2 3
Final Answer The solution set for the inequality 4 x 2 − 8 x + 3 ≤ 0 is the interval [ 2 1 , 2 3 ] .
Examples
Quadratic inequalities are useful in various real-world scenarios, such as optimizing the area of a garden. For example, suppose you want to build a rectangular garden with a fixed perimeter, and you want to ensure that the area of the garden is within a certain range. This problem can be modeled using a quadratic inequality, where the variable represents the length or width of the garden. By solving the inequality, you can determine the possible dimensions of the garden that satisfy the area constraint. Another example is in physics, where projectile motion can be described using quadratic equations, and inequalities can be used to determine the range of the projectile.
