Question
\( \left. \begin{array} { l } { 3 x ^ { 2 } - 7 x - 6 } \\ { 4 x ^ { 2 } - 19 x - 5 } \end{array} \right. \)
Ask by Rodriquez Bradley. in South Africa
Feb 19,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
There are no solutions for \( x \) that satisfy both equations simultaneously.
Solution
Solve the system of equations by following steps:
- step0: Solution:
\(\left\{ \begin{array}{l}3x^{2}-7x-6=0\\4x^{2}-19x-5=0\end{array}\right.\)
- step1: Calculate:
\(\left\{ \begin{array}{l}x=3\cup x=-\frac{2}{3}\\x=5\cup x=-\frac{1}{4}\end{array}\right.\)
- step2: Find the intersection:
\(x \in \varnothing \)
The system of equations given is:
1. \( 3x^2 - 7x - 6 = 0 \)
2. \( 4x^2 - 19x - 5 = 0 \)
Upon solving these equations, we find that there are no common solutions for \( x \) in this system, which means:
\[
x \in \varnothing
\]
This indicates that the two equations do not intersect, and thus there are no values of \( x \) that satisfy both equations simultaneously.
Answered by UpStudy AI and reviewed by a Professional Tutor
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Simplify this solution The Deep Dive
The first expression, \(3x^2 - 7x - 6\), can be factored to find its roots using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 3\), \(b = -7\), and \(c = -6\). By calculating the discriminant, you will find whether the solutions are real and distinct. For the second expression, \(4x^2 - 19x - 5\) also follows the same method. The roots can be determined using the quadratic formula as well. Pay attention to the coefficients; if the discriminant is positive, you’ll discover two real roots, making it possible for the polynomial to cross the x-axis in two places!
