Use the quadratic formula to solve $3 x^2+5 x+2=0$
A) $x=-2 / 3,-1$
B) $x=2,-3$
C) $x=\frac{5 \pm \sqrt{17}}{2}$
D) $x=\frac{-5 \pm \sqrt{19}}{6}$
Answer (1)
Identify the coefficients: a = 3 , b = 5 , c = 2 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c .
Calculate the discriminant: b 2 − 4 a c = 1 .
Find the roots: x = − 3 2 , − 1 . The final answer is x = − 3 2 , − 1 .
Explanation
Problem Analysis We are given the quadratic equation 3 x 2 + 5 x + 2 = 0 . Our goal is to solve for x using the quadratic formula.
Quadratic Formula The quadratic formula is given by: x = 2 a − b ± b 2 − 4 a c where a , b , and c are the coefficients of the quadratic equation a x 2 + b x + c = 0 . In our case, a = 3 , b = 5 , and c = 2 .
Substitution Now, substitute the values of a , b , and c into the quadratic formula: x = 2 ( 3 ) − 5 ± 5 2 − 4 ( 3 ) ( 2 )
Calculating the Discriminant Calculate the discriminant (the term inside the square root): b 2 − 4 a c = 5 2 − 4 ( 3 ) ( 2 ) = 25 − 24 = 1
Simplifying the Formula Substitute the discriminant back into the quadratic formula: x = 2 ( 3 ) − 5 ± 1 = 6 − 5 ± 1
Finding the Roots Now, we find the two possible values of x :
x 1 = 6 − 5 + 1 = 6 − 4 = − 3 2 x 2 = 6 − 5 − 1 = 6 − 6 = − 1
Final Answer Therefore, the solutions are x = − 3 2 and x = − 1 . Comparing these solutions with the given options, we see that option A matches our result.
Examples
Quadratic equations are incredibly useful in various real-world applications. For instance, engineers use them to design bridges and calculate the trajectory of projectiles. Imagine you're launching a rocket; the height of the rocket over time can be modeled by a quadratic equation. By solving this equation, you can determine when the rocket will reach its maximum height or when it will land. This is crucial for ensuring the rocket hits its intended target safely and accurately. Similarly, quadratic equations help optimize designs in construction and physics.
