What is the discriminant of [tex]3 x^2-10 x=-2[/tex]?
Answer (1)
Rewrite the equation in standard form: 3 x 2 − 10 x + 2 = 0 .
Identify the coefficients: a = 3 , b = − 10 , c = 2 .
Calculate the discriminant: D = b 2 − 4 a c = ( − 10 ) 2 − 4 ( 3 ) ( 2 ) = 100 − 24 = 76 .
The discriminant is 76 .
Explanation
Rewrite the equation We are given the quadratic equation 3 x 2 − 10 x = − 2 . To find the discriminant, we first need to rewrite the equation in the standard form a x 2 + b x + c = 0 .
Identify coefficients Adding 2 to both sides of the equation, we get 3 x 2 − 10 x + 2 = 0 . Now we can identify the coefficients: a = 3 , b = − 10 , and c = 2 .
Calculate the discriminant The discriminant D is given by the formula D = b 2 − 4 a c . Substituting the values of a , b , and c , we have D = ( − 10 ) 2 − 4 ( 3 ) ( 2 ) = 100 − 24 = 76.
State the answer Therefore, the discriminant of the quadratic equation 3 x 2 − 10 x = − 2 is 76.
Examples
Understanding the discriminant is crucial in various real-world applications. For instance, in engineering, when designing a bridge, the discriminant can help determine the stability of the structure under different load conditions. By analyzing the discriminant of a quadratic equation that models the bridge's behavior, engineers can predict whether the bridge will remain stable or collapse under stress. Similarly, in physics, the discriminant can be used to analyze the motion of a projectile, determining whether it will hit a target or not. These examples highlight the practical significance of the discriminant in solving real-world problems.
