The discriminant of the equation [tex]$3 x^2+6 x=-3$[/tex] indicates
A. one repeated root
B. two distinct, irrational real roots
C. two distinct, rational roots
D. no real roots
Answer (2)
Rewrite the given equation in standard form: 3 x 2 + 6 x + 3 = 0 .
Identify the coefficients: a = 3 , b = 6 , c = 3 .
Calculate the discriminant: D = b 2 − 4 a c = 6 2 − 4 ( 3 ) ( 3 ) = 0 .
Since D = 0 , the equation has one repeated real root. one repeated root
Explanation
Problem Setup We are given the quadratic equation 3 x 2 + 6 x = − 3 . Our goal is to determine the nature of its roots by analyzing the discriminant.
Standard Form First, we rewrite the equation in the standard form a x 2 + b x + c = 0 . Adding 3 to both sides, we get 3 x 2 + 6 x + 3 = 0 .
Identifying Coefficients Now, we identify the coefficients: a = 3 , b = 6 , and c = 3 .
Calculating the Discriminant Next, we calculate the discriminant D using the formula D = b 2 − 4 a c . Substituting the values, we have D = 6 2 − 4 × 3 × 3 = 36 − 36 = 0.
Interpreting the Discriminant Since the discriminant D = 0 , the quadratic equation has one repeated real root.
Conclusion Therefore, the discriminant of the equation 3 x 2 + 6 x = − 3 indicates one repeated root.
Examples
Understanding the discriminant helps us predict the type of solutions we'll get when solving quadratic equations, which are used in various fields like physics, engineering, and economics. For example, when designing a bridge, engineers use quadratic equations to model the parabolic shape of the bridge's arch. The discriminant helps determine if the arch will intersect the ground at one point (one repeated root), two points (two distinct roots), or not at all (no real roots), ensuring the bridge's stability and safety. In economics, quadratic equations can model cost and revenue functions, and the discriminant can help determine if a company will break even (one repeated root), make a profit (two distinct roots), or incur a loss (no real roots).
By rewriting the quadratic equation and calculating the discriminant, we found that D = 0 . This indicates that the equation has one repeated root. Hence, the correct choice is A: one repeated root.
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