Question
18) \( 3 r^{2}-4 r-4 \)
Ask by Page Riley. in the United States
Mar 12,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\(3r^2 - 4r - 4\) factors to \((3r + 2)(r - 2)\).
Solution
1. Write the quadratic expression:
\[
3r^2 - 4r - 4
\]
2. Identify the coefficients: \( a = 3 \), \( b = -4 \), \( c = -4 \).
3. Compute the product \( ac \):
\[
ac = 3 \times (-4) = -12
\]
4. Find two numbers that multiply to \(-12\) and add to \(b = -4\). These numbers are \(2\) and \(-6\) because:
\[
2 \times (-6) = -12 \quad \text{and} \quad 2 + (-6) = -4
\]
5. Rewrite the middle term using these two numbers:
\[
3r^2 + 2r - 6r - 4
\]
6. Factor by grouping:
- Group the first two terms and the last two terms:
\[
(3r^2 + 2r) + (-6r - 4)
\]
- Factor out the greatest common factor from each group:
\[
r(3r + 2) - 2(3r + 2)
\]
7. Factor out the common factor \((3r + 2)\):
\[
(3r + 2)(r - 2)
\]
Thus, the factored form of \(3r^2 - 4r - 4\) is:
\[
(3r + 2)(r - 2)
\]
Answered by UpStudy AI and reviewed by a Professional Tutor
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Simplify this solution Extra Insights
To tackle the quadratic equation \( 3r^2 - 4r - 4 \), you can use the quadratic formula, which is \( r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, \( a = 3 \), \( b = -4 \), and \( c = -4 \). Plugging these values into the formula lets you find the roots of the equation, allowing you to solve for \( r \). When working with quadratics, always check to make sure you simplify the discriminant (the part under the square root) accurately. A common mistake is miscalculating \( b^2 - 4ac \), so double-check your arithmetic! After you find the roots, it's a good practice to substitute them back into the original equation to verify they truly satisfy it. Happy solving!
