Solve $9 x^2+12 x-24=0$ using the quadratic formula.
A) $x=\frac{9 \pm \sqrt{613}}{2}$
B) $x=4,-7$
C) $x=\frac{-2 \pm 2 \sqrt{7}}{3}$
D) $x=7,-1$
Answer (1)
Identify the coefficients: a = 9 , b = 12 , and c = − 24 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c .
Substitute the values: x = 2 ( 9 ) − 12 ± 1 2 2 − 4 ( 9 ) ( − 24 ) = 18 − 12 ± 1008 .
Simplify the expression: x = 3 − 2 ± 2 7 .
x = 3 − 2 ± 2 7
Explanation
Identifying Coefficients and the Quadratic Formula We are given the quadratic equation 9 x 2 + 12 x − 24 = 0 and asked to solve it using the quadratic formula. The quadratic formula is a general method for finding the solutions (roots) of any quadratic equation in the form a x 2 + b x + c = 0 , where a , b , and c are constants. The formula is given by:
x = 2 a − b ± b 2 − 4 a c
In our equation, we can identify the coefficients as a = 9 , b = 12 , and c = − 24 . We will substitute these values into the quadratic formula to find the solutions for x .
Substituting Values into the Formula Now, we substitute the values of a , b , and c into the quadratic formula:
x = 2 ( 9 ) − 12 ± 1 2 2 − 4 ( 9 ) ( − 24 )
Next, we simplify the expression under the square root:
1 2 2 − 4 ( 9 ) ( − 24 ) = 144 + 864 = 1008
So, we have:
x = 18 − 12 ± 1008
Simplifying the Expression We can simplify the square root by finding the largest perfect square that divides 1008. We have 1008 = 144 × 7 , so 1008 = 144 × 7 = 144 × 7 = 12 7 .
Substituting this back into the expression for x , we get:
x = 18 − 12 ± 12 7
Now, we can simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 6:
x = 18 ÷ 6 − 12 ÷ 6 ± 12 7 ÷ 6 = 3 − 2 ± 2 7
Final Answer Therefore, the solutions for x are:
x = 3 − 2 + 2 7 and x = 3 − 2 − 2 7
Comparing our solutions to the given options, we see that option C matches our result.
Conclusion Thus, the solution to the quadratic equation 9 x 2 + 12 x − 24 = 0 using the quadratic formula is:
x = 3 − 2 ± 2 7
Therefore, the correct answer is C.
Examples
The quadratic formula is a fundamental tool in algebra and has numerous applications in real-world scenarios. For example, engineers use quadratic equations to model the trajectory of projectiles, such as rockets or balls. By knowing the initial velocity and launch angle, they can predict the range and maximum height of the projectile. Similarly, in physics, quadratic equations are used to describe the motion of objects under constant acceleration, such as a car accelerating from rest. Understanding how to solve quadratic equations allows us to analyze and predict the behavior of these systems, making it an essential skill in various scientific and engineering fields.
