Solve the equation [tex]$3 x^2+24 x-24=0$[/tex]

Divide the equation by 3 to simplify: x 2 + 8 x − 8 = 0 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c ​ ​ , where a = 1 , b = 8 , c = − 8 .
Simplify the expression: x = 2 − 8 ± 96 ​ ​ = 2 − 8 ± 4 6 ​ ​ .
The solutions are: x = − 4 ± 2 6 ​ . Thus, x = − 4 + 2 6 ​ and x = − 4 − 2 6 ​ .
x = − 4 ± 2 6 ​ ​

Explanation

Understanding the Problem We are given the quadratic equation 3 x 2 + 24 x − 24 = 0 and asked to solve for x . This means we need to find the values of x that satisfy this equation.

Simplifying the Equation To make the equation easier to work with, we can divide all terms by 3: 3 3 x 2 ​ + 3 24 x ​ − 3 24 ​ = 3 0 ​ x 2 + 8 x − 8 = 0

Applying the Quadratic Formula Now we can use the quadratic formula to solve for x . The quadratic formula is given by: x = 2 a − b ± b 2 − 4 a c ​ ​ where a = 1 , b = 8 , and c = − 8 in our simplified equation.

Substituting Values Substitute the values of a , b , and c into the quadratic formula: x = 2 ( 1 ) − 8 ± 8 2 − 4 ( 1 ) ( − 8 ) ​ ​

Simplifying the Discriminant Simplify the expression under the square root: 8 2 − 4 ( 1 ) ( − 8 ) = 64 + 32 = 96

Simplifying the Square Root Simplify the square root: 96 ​ = 16 ⋅ 6 ​ = 4 6 ​

Substituting Back Substitute the simplified square root back into the quadratic formula: x = 2 − 8 ± 4 6 ​ ​

Final Simplification Divide both terms in the numerator by 2: x = − 4 ± 2 6 ​

The Solutions Therefore, the solutions are x = − 4 + 2 6 ​ and x = − 4 − 2 6 ​ .


Examples
Quadratic equations are used in various real-life applications, such as calculating the trajectory of a projectile, determining the dimensions of a rectangular area given its perimeter and area, and modeling the growth or decay of populations. For example, if you're launching a rocket, you can use a quadratic equation to predict its path, ensuring it reaches its target safely. Similarly, architects use quadratic equations to design structures, ensuring stability and optimal use of materials. Understanding how to solve quadratic equations is essential for solving many practical problems in science, engineering, and everyday life.

Answered by GinnyAnswer | 2025-07-03

To solve the equation 3 x 2 + 24 x − 24 = 0 , we first simplify it to x 2 + 8 x − 8 = 0 by dividing by 3. Then we apply the quadratic formula and find the solutions to be x = − 4 ± 2 6 ​ .
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Answered by Anonymous | 2025-07-04