In the equation $3 x^2-10 x=8$, which of the following values for $x$ will make the equation true?
A. -4
B. -3
C. 3
D. 4
Answer (2)
Rewrite the equation in standard form: 3 x 2 − 10 x − 8 = 0 .
Substitute each option into the equation.
Check if the result is equal to 0.
The value that satisfies the equation is 4 .
Explanation
Understanding the Problem We are given the quadratic equation 3 x 2 − 10 x = 8 and asked to find which of the given values for x makes the equation true. The given options are A. -4, B. -3, C. 3, D. 4.
Rewriting the Equation First, we rewrite the equation in the standard quadratic form by subtracting 8 from both sides:
3 x 2 − 10 x − 8 = 0
Testing the Options Now, we substitute each of the given values of x into the equation 3 x 2 − 10 x − 8 and check if the result is equal to 0.
A. If x = − 4 , then
3 ( − 4 ) 2 − 10 ( − 4 ) − 8 = 3 ( 16 ) + 40 − 8 = 48 + 40 − 8 = 80 e q 0 .
B. If x = − 3 , then
3 ( − 3 ) 2 − 10 ( − 3 ) − 8 = 3 ( 9 ) + 30 − 8 = 27 + 30 − 8 = 49 e q 0 .
C. If x = 3 , then
3 ( 3 ) 2 − 10 ( 3 ) − 8 = 3 ( 9 ) − 30 − 8 = 27 − 30 − 8 = − 11 e q 0 .
D. If x = 4 , then
3 ( 4 ) 2 − 10 ( 4 ) − 8 = 3 ( 16 ) − 40 − 8 = 48 − 40 − 8 = 0 .
Finding the Solution Therefore, the value of x that satisfies the equation is 4.
Examples
Quadratic equations are used in various real-life scenarios, such as calculating the trajectory of a projectile, determining the dimensions of a rectangular area given its area and a relationship between its sides, or modeling the growth of a population. For example, if you are designing a bridge, you might use a quadratic equation to model the curve of an arch, ensuring it can withstand the necessary loads. Similarly, in business, quadratic equations can help model profit and loss scenarios to optimize pricing strategies.
The value of x that satisfies the equation 3 x 2 − 10 x = 8 is 4, as confirmed by substituting the value into the equation and finding that it makes the left-hand side equal to zero.
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