To solve the problem, we need to follow these steps:
1. Form the Equation:
Let [tex]\( x \)[/tex] be the unknown number. According to the problem, when the product of 6 and the square of [tex]\( x \)[/tex] is increased by 5 times [tex]\( x \)[/tex], the result is 4. This translates to the equation:
[tex]\[
6x^2 + 5x = 4
\][/tex]
2. Rearrange the Equation:
To solve the equation, we should rearrange it to a standard quadratic form:
[tex]\[
6x^2 + 5x - 4 = 0
\][/tex]
3. Solve the Quadratic Equation:
We solve the quadratic equation [tex]\( 6x^2 + 5x - 4 = 0 \)[/tex]. The solutions to this equation are the roots of the quadratic equation.
4. Identify the Possible Values:
The solutions to the equation are:
[tex]\[
x = -\frac{4}{3} \quad \text{and} \quad x = \frac{1}{2}
\][/tex]
5. Verify against Given Options:
We need to determine which of the given options align with the solutions obtained:
- [tex]\( -\frac{4}{3} \)[/tex] is a correct solution.
- [tex]\( \frac{1}{2} \)[/tex] is a correct solution.
- [tex]\( -\frac{3}{4} \)[/tex] is not a solution.
- 2 is not a solution.
Conclusion:
The values that the number could be are:
[tex]\[
-\frac{4}{3}, \quad \frac{1}{2}
\][/tex]
Thus, selecting all the correct values from the given options, we have:
[tex]\[
-\frac{4}{3}, \quad \frac{1}{2}
\][/tex]
