Use the discriminant to determine the nature of the solutions to the quadratic equations:
a) [tex]4 x^2+5 x+7[/tex]
b) [tex]3 x^2-6 x+3[/tex]
c) [tex]x^2+6 x-2[/tex]
Answer (2)
Calculate the discriminant D = b 2 − 4 a c for each quadratic equation.
For 4 x 2 + 5 x + 7 , D = − 87 , indicating two complex solutions.
For 3 x 2 − 6 x + 3 , D = 0 , indicating one real solution (a repeated root).
For x 2 + 6 x − 2 , D = 44 , indicating two distinct real solutions. The nature of the solutions is determined by the sign of the discriminant: negative for complex, zero for one real, and positive for two distinct real solutions. Tw o co m pl e x so l u t i o n s , O n e re a l so l u t i o n , Tw o d i s t in c t re a l so l u t i o n s
Explanation
Understanding the Discriminant We are given three quadratic equations and we need to determine the nature of their solutions using the discriminant. The discriminant, denoted as D , is calculated using the formula D = b 2 − 4 a c , where a , b , and c are the coefficients of the quadratic equation a x 2 + b x + c = 0 . The nature of the solutions depends on the value of D :
If 0"> D > 0 , the equation has two distinct real solutions.
If D = 0 , the equation has one real solution (a repeated root).
If D < 0 , the equation has two complex solutions.
Calculating Discriminant for a) a) For the quadratic equation 4 x 2 + 5 x + 7 = 0 , we have a = 4 , b = 5 , and c = 7 . The discriminant is: D = b 2 − 4 a c = 5 2 − 4 ( 4 ) ( 7 ) = 25 − 112 = − 87 Since D = − 87 < 0 , the equation has two complex solutions.
Calculating Discriminant for b) b) For the quadratic equation 3 x 2 − 6 x + 3 = 0 , we have a = 3 , b = − 6 , and c = 3 . The discriminant is: D = b 2 − 4 a c = ( − 6 ) 2 − 4 ( 3 ) ( 3 ) = 36 − 36 = 0 Since D = 0 , the equation has one real solution (a repeated root).
Calculating Discriminant for c) c) For the quadratic equation x 2 + 6 x − 2 = 0 , we have a = 1 , b = 6 , and c = − 2 . The discriminant is: D = b 2 − 4 a c = 6 2 − 4 ( 1 ) ( − 2 ) = 36 + 8 = 44 Since 0"> D = 44 > 0 , the equation has two distinct real solutions.
Final Answer In summary:
a) 4 x 2 + 5 x + 7 has two complex solutions. b) 3 x 2 − 6 x + 3 has one real solution (a repeated root). c) x 2 + 6 x − 2 has two distinct real solutions.
Examples
Understanding the discriminant is crucial in various fields, such as physics and engineering, where quadratic equations frequently arise. For instance, when analyzing the motion of a projectile, the discriminant can determine whether the projectile will hit a target (two real solutions), graze it (one real solution), or miss it entirely (complex solutions). Similarly, in electrical engineering, the discriminant helps determine the stability of a circuit. By analyzing the discriminant, engineers can predict the behavior of systems and design them to meet specific requirements.
The discriminant is used to determine the nature of solutions of quadratic equations. For the equations provided, 4 x 2 + 5 x + 7 has two complex solutions, 3 x 2 − 6 x + 3 has one real solution, and x 2 + 6 x − 2 has two distinct real solutions.
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