Question
SUBMIT ANSWER 4. [0/1 Points] DETAILS MY NOTES GHCOLALG13 Consider the following equation. \[ 7 x^{2}+24 x-16=0 \] Factor the expression \( 7 x^{2}+24 x-16 \)
Ask by Patel Riley. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The factored form of \( 7x^{2} + 24x - 16 \) is \( (7x - 4)(x + 4) \).
Solution
To factor the quadratic expression \( 7x^{2} + 24x - 16 \), follow these steps:
1. **Identify the coefficients:**
- \( a = 7 \)
- \( b = 24 \)
- \( c = -16 \)
2. **Find two numbers that multiply to \( a \times c = 7 \times (-16) = -112 \) and add up to \( b = 24 \):**
- The numbers **28** and **-4** satisfy this because:
- \( 28 \times (-4) = -112 \)
- \( 28 + (-4) = 24 \)
3. **Rewrite the middle term using these numbers:**
\[
7x^{2} + 28x - 4x - 16
\]
4. **Factor by grouping:**
\[
(7x^{2} + 28x) + (-4x - 16) = 7x(x + 4) - 4(x + 4)
\]
5. **Factor out the common binomial factor \((x + 4)\):**
\[
(7x - 4)(x + 4)
\]
**Final Factored Form:**
\[
7x^{2} + 24x - 16 = (7x - 4)(x + 4)
\]
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Simplify this solution Beyond the Answer
To factor the expression \( 7 x^{2}+24 x-16 \), we want to find two numbers that multiply to \( 7 \times -16 = -112 \) and add up to \( 24 \). The numbers \( 28 \) and \( -4 \) work perfectly since \( 28 - 4 = 24 \) and \( 28 \times -4 = -112 \). Now, rewrite the equation as: \[ 7x^{2} + 28x - 4x - 16 = 0. \] Next, group the terms: \[ (7x^{2} + 28x) + (-4x - 16) = 0, \] which can be factored as: \[ 7x(x + 4) - 4(x + 4) = 0. \] This gives us: \[ (7x - 4)(x + 4) = 0. \] To solve the equation, set each factor to zero: 1. \( 7x - 4 = 0 \) leads to \( x = \frac{4}{7} \). 2. \( x + 4 = 0 \) leads to \( x = -4 \). So, the factored form is \( (7x - 4)(x + 4) = 0 \).
