Factor `8x^{2}+16x-10`completely. Which of the following expressions is a factor?

Question

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Answer 1

To factor the quadratic expression \(8x^{2}+16x-10\), we can use the following steps:

1. Identify the coefficients in our quadratic expression \(ax^2 + bx + c\):

   • \(a = 8\)
   • \(b = 16\)
   • \(c = -10\)

2. Factor out the greatest common factor (GCF): The GCF of the coefficients \(8, 16,\) and \(-10\) is \(2\). We factor \(2\) out of the entire expression:

   \[ 8x^2 + 16x - 10 = 2(4x^2 + 8x - 5) \]

3. Now, we need to factor the quadratic \(4x^2 + 8x - 5\). We will use the method of factoring by grouping or the AC method.

   • To apply the AC method, we compute \(A \times C\), where \(A = 4\) and \(C = -5\): \[ AC = 4 \times (-5) = -20 \]

   • We need to find two numbers that multiply to \(-20\) and add up to \(B = 8\). The numbers \(10\) and \(-2\) meet this criterion because: \[ 10 \times (-2) = -20 \quad \text{and} \quad 10 + (-2) = 8 \]

4. Rewrite the expression using these two numbers: \[ 4x^2 + 10x - 2x - 5 \]

5. Group the terms: \[ (4x^2 + 10x) + (-2x - 5) \]

6. Factor by grouping:

   • From the first group, we can factor out \(2x\): \[ 2x(2x + 5) \]
   • From the second group, we can factor out \(-1\): \[ -1(2x + 5) \]

Now we can combine both parts: \[ 2x(2x + 5) - 1(2x + 5) \] \[ = (2x - 1)(2x + 5) \]

7. Combine everything: Don't forget the \(2\) we factored out initially: \[ 8x^2 + 16x - 10 = 2(2x - 1)(2x + 5) \]

8. Conclusion: The completely factored form of \(8x^{2}+16x-10\) is: \[ 2(2x - 1)(2x + 5) \]

One of the factors of the original expression is \(2x - 1\) or \(2x + 5\). So, you can choose either as a valid factor.