Solve $2 x^2-3 x=12$ using the quadratic formula.
A) $x=\frac{3}{4}+\frac{\sqrt{87}}{4}, x=\frac{3}{4}-\frac{\sqrt{87}}{4}$
B) $x=\frac{3}{4}+\frac{\sqrt{87}}{4} i, x=\frac{3}{4}-\frac{\sqrt{87}}{4} i$
C) $x=\frac{3}{4}+\frac{\sqrt{105}}{4}, x=\frac{3}{4}-\frac{\sqrt{105}}{4}$
D) $x=\frac{3}{4}+\frac{\sqrt{105}}{4} i, x=\frac{3}{4}-\frac{\sqrt{105}}{4} i$
Answer (1)
Rewrite the equation in standard form: 2 x 2 − 3 x − 12 = 0 .
Identify the coefficients: a = 2 , b = − 3 , c = − 12 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c = 4 3 ± 105 .
The solutions are x = 4 3 + 4 105 and x = 4 3 − 4 105 , so the answer is x = 4 3 + 4 105 , x = 4 3 − 4 105 .
Explanation
Understanding the Problem We are given the quadratic equation 2 x 2 − 3 x = 12 . Our goal is to solve this equation using the quadratic formula. The quadratic formula is a general method for finding the solutions (also called roots) of any quadratic equation, which is an equation of the form a x 2 + b x + c = 0 , where a , b , and c are constants.
Rewriting the Equation First, we need to rewrite the given equation in the standard form a x 2 + b x + c = 0 . To do this, we subtract 12 from both sides of the equation:
2 x 2 − 3 x − 12 = 0
Identifying Coefficients Now we can identify the coefficients a , b , and c . In this case, we have:
a = 2 b = − 3 c = − 12
Stating the Quadratic Formula The quadratic formula is given by:
x = 2 a − b ± b 2 − 4 a c
Substituting Values Now we substitute the values of a , b , and c into the quadratic formula:
x = 2 ( 2 ) − ( − 3 ) ± ( − 3 ) 2 − 4 ( 2 ) ( − 12 )
Simplifying the Expression Simplify the expression:
x = 4 3 ± 9 + 96
x = 4 3 ± 105
Finding the Solutions So the two solutions for x are:
x = 4 3 + 105
x = 4 3 − 105
Comparing with Options Therefore, the solutions are x = 4 3 + 4 105 and x = 4 3 − 4 105 . Comparing these solutions with the given options, we see that option C matches our result.
Examples
The quadratic formula is incredibly useful in various real-world scenarios. For instance, engineers use it to calculate the trajectory of projectiles, ensuring accuracy in designs and safety measures. Similarly, financial analysts apply it to model investment growth and predict potential returns, helping investors make informed decisions. Even in sports, understanding quadratic equations can optimize performance, such as determining the best angle to kick a ball for maximum distance. These applications highlight the practical significance of mastering the quadratic formula.
