Factor. [tex]25 b^2+29 b+9[/tex]
Answer (2)
The quadratic expression 25 b 2 + 29 b + 9 cannot be factored using real coefficients because its discriminant is negative ( − 59 ). This indicates the absence of real roots. Hence, it cannot be factored into two binomials with real numbers.
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Calculate the discriminant: Δ = b 2 − 4 a c = 2 9 2 − 4 × 25 × 9 = − 59 .
Since the discriminant is negative, the quadratic expression cannot be factored using real numbers.
The quadratic expression 25 b 2 + 29 b + 9 is not factorable with real coefficients.
The final answer is that the expression is not factorable. cannot be factored
Explanation
Problem Analysis We are asked to factor the quadratic expression 25 b 2 + 29 b + 9 . To determine if this quadratic can be factored, we can calculate the discriminant.
Calculating the Discriminant The discriminant, denoted as Δ , is given by the formula Δ = b 2 − 4 a c , where a , b , and c are the coefficients of the quadratic expression a x 2 + b x + c . In our case, a = 25 , b = 29 , and c = 9 .
Determining Factorability Substituting these values into the discriminant formula, we get: Δ = 2 9 2 − 4 ( 25 ) ( 9 ) = 841 − 900 = − 59 Since the discriminant is negative, the quadratic expression 25 b 2 + 29 b + 9 cannot be factored into two binomials with real coefficients.
Conclusion Therefore, the quadratic expression 25 b 2 + 29 b + 9 cannot be factored using real numbers.
Examples
Factoring quadratic expressions is a fundamental skill in algebra with numerous real-world applications. For instance, engineers use factoring to analyze the stability of structures, economists use it to model supply and demand curves, and computer scientists use it to optimize algorithms. Imagine you are designing a bridge and need to ensure it can withstand certain loads. By expressing the load as a quadratic equation and factoring it, you can identify critical points where the bridge might be vulnerable. This allows you to reinforce those areas and ensure the bridge's safety and stability.
