Solve the quadratic equation using any method. Give exact answers - no decimals.

[tex]2 w^2+3 w-10=0[/tex]

Identify the coefficients: a = 2 , b = 3 , and c = − 10 .
Apply the quadratic formula: w = 2 a − b ± b 2 − 4 a c ​ ​ .
Substitute the values and simplify: w = 2 ( 2 ) − 3 ± 3 2 − 4 ( 2 ) ( − 10 ) ​ ​ = 4 − 3 ± 89 ​ ​ .
State the two exact solutions: w = 4 − 3 ± 89 ​ ​ ​ .

Explanation

Understanding the Problem We are given the quadratic equation 2 w 2 + 3 w − 10 = 0 . Our goal is to find the exact solutions for w , without using decimals. We can use the quadratic formula to solve for w .

Applying the Quadratic Formula The quadratic formula is given by w = 2 a − b ± b 2 − 4 a c ​ ​ where a = 2 , b = 3 , and c = − 10 .

Substituting Values Substitute the values of a , b , and c into the quadratic formula: w = 2 ( 2 ) − 3 ± 3 2 − 4 ( 2 ) ( − 10 ) ​ ​

Simplifying the Discriminant Simplify the expression under the square root: 3 2 − 4 ( 2 ) ( − 10 ) = 9 + 80 = 89

Finding the Solutions Therefore, the solutions are: w = 4 − 3 ± 89 ​ ​

Exact Solutions The two solutions are: w = 4 − 3 + 89 ​ ​ and w = 4 − 3 − 89 ​ ​ These are the exact solutions for w .


Examples
Quadratic equations are incredibly useful in various real-world scenarios. For instance, imagine you're designing a rectangular garden and you know the total area and the relationship between the length and width. You can use a quadratic equation to determine the exact dimensions of the garden. Similarly, in physics, quadratic equations are used to model projectile motion, helping to calculate the trajectory of a ball thrown in the air or the path of a rocket. Understanding how to solve these equations allows you to make accurate predictions and informed decisions in these and many other practical situations.

Answered by GinnyAnswer | 2025-07-07