What is the discriminant of the polynomial below?
[tex]2 x^2+5 x-8[/tex]
A. 89
B. -39
C. 31
D. -59
Answer (2)
Identify the coefficients of the quadratic polynomial: a = 2 , b = 5 , and c = − 8 .
Apply the discriminant formula: Δ = b 2 − 4 a c .
Substitute the values into the formula: Δ = ( 5 ) 2 − 4 ( 2 ) ( − 8 ) .
Calculate the discriminant: Δ = 89 . The final answer is 89 .
Explanation
Understanding the Problem We are asked to find the discriminant of the quadratic polynomial 2 x 2 + 5 x − 8 . The discriminant is a value that helps us determine the nature of the roots of the quadratic equation.
Identifying Coefficients The general form of a quadratic polynomial is a x 2 + b x + c , where a , b , and c are coefficients. In our case, we have a = 2 , b = 5 , and c = − 8 .
Stating the Discriminant Formula The discriminant, denoted as Δ , is given by the formula Δ = b 2 − 4 a c . We will substitute the values of a , b , and c into this formula.
Calculating the Discriminant Substituting the values, we get: Δ = ( 5 ) 2 − 4 ( 2 ) ( − 8 ) Δ = 25 − ( − 64 ) Δ = 25 + 64 Δ = 89
Final Answer Therefore, the discriminant of the polynomial 2 x 2 + 5 x − 8 is 89.
Examples
Understanding the discriminant is useful in various real-world applications. For example, when designing a bridge, engineers need to ensure that the structure can withstand different loads and stresses. The discriminant can help determine the stability of the bridge under different conditions. Similarly, in physics, the discriminant can be used to analyze the motion of objects and determine whether they will reach a certain point or not. In general, the discriminant helps to analyze the nature of solutions in any quadratic model.
The discriminant of the polynomial 2 x 2 + 5 x − 8 is 89 . This is calculated using the formula Δ = b 2 − 4 a c , leading to two distinct real roots for the equation. Therefore, the correct option is A. 89.
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