Factor completely $-4 x^2+16 x-24$
A. $-1(x^2-16 x+24)$
B. $-4(x^2-4 x+6)$
C. $-4 x(x^2-4 x+6)$
D. $-4(x^2+4 x-6)$
Answer (2)
Factor out the greatest common factor (GCF) -4: − 4 ( x 2 − 4 x + 6 )
Check if the quadratic x 2 − 4 x + 6 can be factored further using integer coefficients. It cannot.
Use the quadratic formula to find the roots of x 2 − 4 x + 6 = 0 . The roots are complex, so the quadratic cannot be factored using real numbers.
The completely factored form of the original expression is: − 4 ( x 2 − 4 x + 6 )
Explanation
Problem Analysis We are given the quadratic expression − 4 x 2 + 16 x − 24 and asked to factor it completely.
Factoring out the GCF First, we factor out the greatest common factor (GCF) from all terms. The GCF of -4, 16, and -24 is -4. Factoring out -4, we get: − 4 ( x 2 − 4 x + 6 )
Checking for Further Factoring with Integer Coefficients Now, we need to check if the quadratic expression inside the parentheses, x 2 − 4 x + 6 , can be factored further using integer coefficients. To do this, we look for two numbers that multiply to 6 and add up to -4. The pairs of factors of 6 are (1, 6) and (2, 3). Neither of these pairs can be combined to give a sum of -4. Therefore, the quadratic x 2 − 4 x + 6 cannot be factored using integer coefficients.
Finding the Roots Using the Quadratic Formula To determine if x 2 − 4 x + 6 can be factored using real numbers, we can find the roots of the quadratic equation x 2 − 4 x + 6 = 0 using the quadratic formula: x = 2 a − b ± b 2 − 4 a c In this case, a = 1 , b = − 4 , and c = 6 . Plugging these values into the quadratic formula, we get: x = 2 ( 1 ) − ( − 4 ) ± ( − 4 ) 2 − 4 ( 1 ) ( 6 ) = 2 4 ± 16 − 24 = 2 4 ± − 8 Since the discriminant (the value inside the square root) is negative, the roots are complex numbers. Therefore, the quadratic x 2 − 4 x + 6 cannot be factored into linear factors with real coefficients. The roots are 2 + i 2 and 2 − i 2 .
Final Factored Form Since the quadratic x 2 − 4 x + 6 cannot be factored further using real numbers, the completely factored form of the original expression is: − 4 ( x 2 − 4 x + 6 )
Examples
Factoring quadratics is a fundamental skill in algebra and is used in many real-world applications. For example, 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. Factoring helps simplify complex expressions and solve equations, making it an essential tool in various fields.
The expression − 4 x 2 + 16 x − 24 factors to − 4 ( x 2 − 4 x + 6 ) . The quadratic x 2 − 4 x + 6 cannot be factored further using integer coefficients or real numbers because its discriminant is negative. Therefore, the complete factorization is − 4 ( x 2 − 4 x + 6 ) .
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