Given the formula $6 x^2+7 x-8$ find its value when x is equal to:
$
\begin{array}{l}
x=0: \square \\
x=2: \square \\
x=-2: \square
\end{array}$
Answer (1)
Substitute x = 0 into the expression: 6 ( 0 ) 2 + 7 ( 0 ) − 8 = − 8 .
Substitute x = 2 into the expression: 6 ( 2 ) 2 + 7 ( 2 ) − 8 = 30 .
Substitute x = − 2 into the expression: 6 ( − 2 ) 2 + 7 ( − 2 ) − 8 = 2 .
The values are − 8 , 30 , 2 .
Explanation
Problem Analysis We are given the quadratic expression 6 x 2 + 7 x − 8 and we need to evaluate it for x = 0 , x = 2 , and x = − 2 . We will substitute each value of x into the expression and simplify.
Evaluating for x=0 First, let's evaluate the expression for x = 0 :
6 ( 0 ) 2 + 7 ( 0 ) − 8 = 6 ( 0 ) + 0 − 8 = 0 + 0 − 8 = − 8
Evaluating for x=2 Next, let's evaluate the expression for x = 2 :
6 ( 2 ) 2 + 7 ( 2 ) − 8 = 6 ( 4 ) + 14 − 8 = 24 + 14 − 8 = 38 − 8 = 30
Evaluating for x=-2 Finally, let's evaluate the expression for x = − 2 :
6 ( − 2 ) 2 + 7 ( − 2 ) − 8 = 6 ( 4 ) − 14 − 8 = 24 − 14 − 8 = 10 − 8 = 2
Final Answer Therefore, the values of the expression for x = 0 , x = 2 , and x = − 2 are − 8 , 30 , and 2 , respectively.
Examples
Understanding how to evaluate polynomial expressions is crucial in many real-world applications. For instance, engineers use polynomial functions to model the trajectory of a projectile, such as a ball thrown in the air. By substituting different values of time (x) into the polynomial, they can determine the height of the ball at various points in its flight. Similarly, economists use polynomial functions to model cost and revenue curves, allowing them to predict profits at different production levels. Evaluating these functions at specific points helps in making informed decisions and optimizing outcomes.
