Answer :
To determine which quadratic equation can be used to find the thickness of the painting, we will analyze the solutions to each given equation. The given options are:
1. [tex]\( 4x^2 + 22x - 1 = 0 \)[/tex]
2. [tex]\( 4x^2 + 22x + 31 = 0 \)[/tex]
3. [tex]\( x^2 + 11x - 1 = 0 \)[/tex]
4. [tex]\( x^2 + 11x + 31 = 0 \)[/tex]
We need to solve each quadratic equation and interpret the solutions within the context of the problem to determine if any of the solutions are reasonable for the thickness. The solutions to these equations can be found through methods such as the quadratic formula. However, the results are:
1. For the equation [tex]\( 4x^2 + 22x - 1 = 0 \)[/tex]:
[tex]\[ x = -\frac{11}{4} + \frac{5\sqrt{5}}{4}, \quad x = -\frac{11}{4} - \frac{5\sqrt{5}}{4} \][/tex]
2. For the equation [tex]\( 4x^2 + 22x + 31 = 0 \)[/tex]:
[tex]\[ x = -\frac{11}{4} - \frac{\sqrt{3}i}{4}, \quad x = -\frac{11}{4} + \frac{\sqrt{3}i}{4} \][/tex]
3. For the equation [tex]\( x^2 + 11x - 1 = 0 \)[/tex]:
[tex]\[ x = -\frac{11}{2} + \frac{5\sqrt{5}}{2}, \quad x = -\frac{11}{2} - \frac{5\sqrt{5}}{2} \][/tex]
4. For the equation [tex]\( x^2 + 11x + 31 = 0 \)[/tex]:
[tex]\[ x = -\frac{11}{2} - \frac{\sqrt{3}i}{2}, \quad x = -\frac{11}{2} + \frac{\sqrt{3}i}{2} \][/tex]
Now, we can analyze the solutions:
1. The solutions for [tex]\( 4x^2 + 22x - 1 = 0 \)[/tex]:
- [tex]\( x = -\frac{11}{4} + \frac{5\sqrt{5}}{4} \)[/tex]
- [tex]\( x = -\frac{11}{4} - \frac{5\sqrt{5}}{4} \)[/tex]
Both solutions are real numbers. We need to check if they yield a positive thickness:
- [tex]\( -\frac{11}{4} + \frac{5\sqrt{5}}{4} \)[/tex] can be a positive number.
- [tex]\( -\frac{11}{4} - \frac{5\sqrt{5}}{4} \)[/tex] is definitely a negative number.
Only [tex]\( -\frac{11}{4} + \frac{5\sqrt{5}}{4} \)[/tex] is potentially positive.
2. The solutions for [tex]\( 4x^2 + 22x + 31 = 0 \)[/tex]:
- [tex]\( x = -\frac{11}{4} - \frac{\sqrt{3}i}{4} \)[/tex]
- [tex]\( x = -\frac{11}{4} + \frac{\sqrt{3}i}{4} \)[/tex]
These solutions are complex numbers and do not correspond to a physical thickness.
3. The solutions for [tex]\( x^2 + 11x - 1 = 0 \)[/tex]:
- [tex]\( x = -\frac{11}{2} + \frac{5\sqrt{5}}{2} \)[/tex]
- [tex]\( x = -\frac{11}{2} - \frac{5\sqrt{5}}{2} \)[/tex]
Both solutions are real numbers. We need to check if they yield a positive thickness:
- [tex]\( -\frac{11}{2} + \frac{5\sqrt{5}}{2} \)[/tex] can be a positive number.
- [tex]\( -\frac{11}{2} - \frac{5\sqrt{5}}{2} \)[/tex] is definitely a negative number.
Only [tex]\( -\frac{11}{2} + \frac{5\sqrt{5}}{2} \)[/tex] is potentially positive.
4. The solutions for [tex]\( x^2 + 11x + 31 = 0 \)[/tex]:
- [tex]\( x = -\frac{11}{2} - \frac{\sqrt{3}i}{2} \)[/tex]
- [tex]\( x = -\frac{11}{2} + \frac{\sqrt{3}i}{2} \)[/tex]
These solutions are complex numbers and do not correspond to a physical thickness.
Conclusion:
To find a reasonable equation that can potentially yield a positive thickness, we can consider the equations from options (1) and (3) since each provided a real positive solution candidate:
- [tex]\( 4x^2 + 22x - 1 = 0 \)[/tex]
- [tex]\( x^2 + 11x - 1 = 0 \)[/tex]
Thus, either [tex]\( 4x^2 + 22x - 1 = 0 \)[/tex] or [tex]\( x^2 + 11x - 1 = 0 \)[/tex] can be used to determine the thickness of the painting.
1. [tex]\( 4x^2 + 22x - 1 = 0 \)[/tex]
2. [tex]\( 4x^2 + 22x + 31 = 0 \)[/tex]
3. [tex]\( x^2 + 11x - 1 = 0 \)[/tex]
4. [tex]\( x^2 + 11x + 31 = 0 \)[/tex]
We need to solve each quadratic equation and interpret the solutions within the context of the problem to determine if any of the solutions are reasonable for the thickness. The solutions to these equations can be found through methods such as the quadratic formula. However, the results are:
1. For the equation [tex]\( 4x^2 + 22x - 1 = 0 \)[/tex]:
[tex]\[ x = -\frac{11}{4} + \frac{5\sqrt{5}}{4}, \quad x = -\frac{11}{4} - \frac{5\sqrt{5}}{4} \][/tex]
2. For the equation [tex]\( 4x^2 + 22x + 31 = 0 \)[/tex]:
[tex]\[ x = -\frac{11}{4} - \frac{\sqrt{3}i}{4}, \quad x = -\frac{11}{4} + \frac{\sqrt{3}i}{4} \][/tex]
3. For the equation [tex]\( x^2 + 11x - 1 = 0 \)[/tex]:
[tex]\[ x = -\frac{11}{2} + \frac{5\sqrt{5}}{2}, \quad x = -\frac{11}{2} - \frac{5\sqrt{5}}{2} \][/tex]
4. For the equation [tex]\( x^2 + 11x + 31 = 0 \)[/tex]:
[tex]\[ x = -\frac{11}{2} - \frac{\sqrt{3}i}{2}, \quad x = -\frac{11}{2} + \frac{\sqrt{3}i}{2} \][/tex]
Now, we can analyze the solutions:
1. The solutions for [tex]\( 4x^2 + 22x - 1 = 0 \)[/tex]:
- [tex]\( x = -\frac{11}{4} + \frac{5\sqrt{5}}{4} \)[/tex]
- [tex]\( x = -\frac{11}{4} - \frac{5\sqrt{5}}{4} \)[/tex]
Both solutions are real numbers. We need to check if they yield a positive thickness:
- [tex]\( -\frac{11}{4} + \frac{5\sqrt{5}}{4} \)[/tex] can be a positive number.
- [tex]\( -\frac{11}{4} - \frac{5\sqrt{5}}{4} \)[/tex] is definitely a negative number.
Only [tex]\( -\frac{11}{4} + \frac{5\sqrt{5}}{4} \)[/tex] is potentially positive.
2. The solutions for [tex]\( 4x^2 + 22x + 31 = 0 \)[/tex]:
- [tex]\( x = -\frac{11}{4} - \frac{\sqrt{3}i}{4} \)[/tex]
- [tex]\( x = -\frac{11}{4} + \frac{\sqrt{3}i}{4} \)[/tex]
These solutions are complex numbers and do not correspond to a physical thickness.
3. The solutions for [tex]\( x^2 + 11x - 1 = 0 \)[/tex]:
- [tex]\( x = -\frac{11}{2} + \frac{5\sqrt{5}}{2} \)[/tex]
- [tex]\( x = -\frac{11}{2} - \frac{5\sqrt{5}}{2} \)[/tex]
Both solutions are real numbers. We need to check if they yield a positive thickness:
- [tex]\( -\frac{11}{2} + \frac{5\sqrt{5}}{2} \)[/tex] can be a positive number.
- [tex]\( -\frac{11}{2} - \frac{5\sqrt{5}}{2} \)[/tex] is definitely a negative number.
Only [tex]\( -\frac{11}{2} + \frac{5\sqrt{5}}{2} \)[/tex] is potentially positive.
4. The solutions for [tex]\( x^2 + 11x + 31 = 0 \)[/tex]:
- [tex]\( x = -\frac{11}{2} - \frac{\sqrt{3}i}{2} \)[/tex]
- [tex]\( x = -\frac{11}{2} + \frac{\sqrt{3}i}{2} \)[/tex]
These solutions are complex numbers and do not correspond to a physical thickness.
Conclusion:
To find a reasonable equation that can potentially yield a positive thickness, we can consider the equations from options (1) and (3) since each provided a real positive solution candidate:
- [tex]\( 4x^2 + 22x - 1 = 0 \)[/tex]
- [tex]\( x^2 + 11x - 1 = 0 \)[/tex]
Thus, either [tex]\( 4x^2 + 22x - 1 = 0 \)[/tex] or [tex]\( x^2 + 11x - 1 = 0 \)[/tex] can be used to determine the thickness of the painting.
