Use the formula and given information to solve for the indicated variable. Round answers to two decimal places as necessary.
[tex]d=b^2-4 a c[/tex]
Given: [tex]a =-9, b=-12[/tex], and [tex]c =7[/tex].
Calculate d.
Answer (1)
Substitute the given values a = − 9 , b = − 12 , and c = 7 into the formula d = b 2 − 4 a c .
Calculate b 2 = ( − 12 ) 2 = 144 .
Calculate − 4 a c = − 4 ( − 9 ) ( 7 ) = 252 .
Calculate d = 144 + 252 = 396 $.
Explanation
Understanding the Problem We are given the formula d = b 2 − 4 a c and the values a = − 9 , b = − 12 , and c = 7 . Our goal is to calculate the value of d by substituting the given values into the formula.
Substituting the Values First, we substitute the given values into the formula: d = ( − 12 ) 2 − 4 ( − 9 ) ( 7 ) .
Calculating ( − 12 ) 2 Next, we calculate ( − 12 ) 2 , which is ( − 12 ) × ( − 12 ) = 144 . So, d = 144 − 4 ( − 9 ) ( 7 ) .
Calculating − 4 a c Now, we calculate − 4 ( − 9 ) ( 7 ) . We have − 4 × − 9 = 36 , and 36 × 7 = 252 . So, d = 144 + 252 .
Calculating d Finally, we calculate 144 + 252 = 396 . Therefore, d = 396 . Since the problem asks us to round to two decimal places if necessary, and 396 is an integer, we can write it as 396.00.
Final Answer Thus, the value of d is 396.
Examples
The discriminant, calculated using the formula d = b 2 − 4 a c , is a crucial part of the quadratic formula. In real-world applications, the discriminant can tell us about the nature of the solutions to a quadratic equation, such as in physics problems involving projectile motion or in engineering when designing structures. For example, if 0"> d > 0 , there are two distinct real roots, indicating two possible solutions to the problem. If d = 0 , there is exactly one real root, and if d < 0 , there are no real roots, meaning there are no real solutions.
