Answer :
Sure! Let's go through the process of solving each equation one by one:
### Problem 10: [tex]\( 3x^4 - 22x^2 - 45 = 0 \)[/tex]
This equation is quadratic in form. We can use substitution to make it simpler:
1. Let [tex]\( u = x^2 \)[/tex]. Then, the equation becomes [tex]\( 3u^2 - 22u - 45 = 0 \)[/tex].
2. Solve the quadratic equation [tex]\( 3u^2 - 22u - 45 = 0 \)[/tex] using the quadratic formula:
[tex]\[
u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here, [tex]\( a = 3 \)[/tex], [tex]\( b = -22 \)[/tex], and [tex]\( c = -45 \)[/tex].
3. Calculate the discriminant:
[tex]\[
b^2 - 4ac = (-22)^2 - 4 \times 3 \times (-45) = 484 + 540 = 1024
\][/tex]
4. Find the roots for [tex]\( u \)[/tex]:
[tex]\[
u = \frac{22 \pm \sqrt{1024}}{6} = \frac{22 \pm 32}{6}
\][/tex]
This gives us [tex]\( u = 9 \)[/tex] or [tex]\( u = -\frac{5}{3} \)[/tex].
5. Convert back to [tex]\( x \)[/tex] by solving [tex]\( x^2 = u \)[/tex]:
- For [tex]\( x^2 = 9 \)[/tex], [tex]\( x = \pm 3 \)[/tex].
- For [tex]\( x^2 = -\frac{5}{3} \)[/tex], the solutions are complex: [tex]\( x = \pm \frac{\sqrt{15}i}{3} \)[/tex].
### Problem 11: [tex]\( 19x^3 - 38x^2 = 0 \)[/tex]
Factor this equation:
1. Factor out the greatest common factor [tex]\( x^2 \)[/tex]:
[tex]\[
x^2(19x - 38) = 0
\][/tex]
2. Set each factor equal to zero:
- [tex]\( x^2 = 0 \)[/tex] gives [tex]\( x = 0 \)[/tex].
- [tex]\( 19x - 38 = 0 \)[/tex] gives [tex]\( x = 2 \)[/tex].
### Problem 12: [tex]\( x^3 - 9x^2 + 14x = 0 \)[/tex]
Factor this equation:
1. Factor out the greatest common factor [tex]\( x \)[/tex]:
[tex]\[
x(x^2 - 9x + 14) = 0
\][/tex]
2. Set each factor equal to zero:
- [tex]\( x = 0 \)[/tex].
- Solve the quadratic equation [tex]\( x^2 - 9x + 14 = 0 \)[/tex] using factorization or the quadratic formula.
3. Factor [tex]\( x^2 - 9x + 14 \)[/tex] as [tex]\( (x - 2)(x - 7) = 0 \)[/tex], giving solutions [tex]\( x = 2 \)[/tex] and [tex]\( x = 7 \)[/tex].
### Problem 13: [tex]\( x^4 - 625 = 0 \)[/tex]
This is a difference of squares:
1. Factor as [tex]\( (x^2 - 25)(x^2 + 25) = 0 \)[/tex].
2. Further factor [tex]\( (x^2 - 25) \)[/tex] as [tex]\( (x - 5)(x + 5) = 0 \)[/tex].
3. Solve each equation:
- [tex]\( x - 5 = 0 \)[/tex] gives [tex]\( x = 5 \)[/tex].
- [tex]\( x + 5 = 0 \)[/tex] gives [tex]\( x = -5 \)[/tex].
4. For [tex]\( (x^2 + 25) = 0 \)[/tex], solve to find the complex roots: [tex]\( x = \pm 5i \)[/tex].
### Summary of Solutions:
- Problem 10: [tex]\( x = -3, 3, \pm \frac{\sqrt{15}i}{3} \)[/tex]
- Problem 11: [tex]\( x = 0, 2 \)[/tex]
- Problem 12: [tex]\( x = 0, 2, 7 \)[/tex]
- Problem 13: [tex]\( x = -5, 5, \pm 5i \)[/tex]
These are the solutions for each given equation.
### Problem 10: [tex]\( 3x^4 - 22x^2 - 45 = 0 \)[/tex]
This equation is quadratic in form. We can use substitution to make it simpler:
1. Let [tex]\( u = x^2 \)[/tex]. Then, the equation becomes [tex]\( 3u^2 - 22u - 45 = 0 \)[/tex].
2. Solve the quadratic equation [tex]\( 3u^2 - 22u - 45 = 0 \)[/tex] using the quadratic formula:
[tex]\[
u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here, [tex]\( a = 3 \)[/tex], [tex]\( b = -22 \)[/tex], and [tex]\( c = -45 \)[/tex].
3. Calculate the discriminant:
[tex]\[
b^2 - 4ac = (-22)^2 - 4 \times 3 \times (-45) = 484 + 540 = 1024
\][/tex]
4. Find the roots for [tex]\( u \)[/tex]:
[tex]\[
u = \frac{22 \pm \sqrt{1024}}{6} = \frac{22 \pm 32}{6}
\][/tex]
This gives us [tex]\( u = 9 \)[/tex] or [tex]\( u = -\frac{5}{3} \)[/tex].
5. Convert back to [tex]\( x \)[/tex] by solving [tex]\( x^2 = u \)[/tex]:
- For [tex]\( x^2 = 9 \)[/tex], [tex]\( x = \pm 3 \)[/tex].
- For [tex]\( x^2 = -\frac{5}{3} \)[/tex], the solutions are complex: [tex]\( x = \pm \frac{\sqrt{15}i}{3} \)[/tex].
### Problem 11: [tex]\( 19x^3 - 38x^2 = 0 \)[/tex]
Factor this equation:
1. Factor out the greatest common factor [tex]\( x^2 \)[/tex]:
[tex]\[
x^2(19x - 38) = 0
\][/tex]
2. Set each factor equal to zero:
- [tex]\( x^2 = 0 \)[/tex] gives [tex]\( x = 0 \)[/tex].
- [tex]\( 19x - 38 = 0 \)[/tex] gives [tex]\( x = 2 \)[/tex].
### Problem 12: [tex]\( x^3 - 9x^2 + 14x = 0 \)[/tex]
Factor this equation:
1. Factor out the greatest common factor [tex]\( x \)[/tex]:
[tex]\[
x(x^2 - 9x + 14) = 0
\][/tex]
2. Set each factor equal to zero:
- [tex]\( x = 0 \)[/tex].
- Solve the quadratic equation [tex]\( x^2 - 9x + 14 = 0 \)[/tex] using factorization or the quadratic formula.
3. Factor [tex]\( x^2 - 9x + 14 \)[/tex] as [tex]\( (x - 2)(x - 7) = 0 \)[/tex], giving solutions [tex]\( x = 2 \)[/tex] and [tex]\( x = 7 \)[/tex].
### Problem 13: [tex]\( x^4 - 625 = 0 \)[/tex]
This is a difference of squares:
1. Factor as [tex]\( (x^2 - 25)(x^2 + 25) = 0 \)[/tex].
2. Further factor [tex]\( (x^2 - 25) \)[/tex] as [tex]\( (x - 5)(x + 5) = 0 \)[/tex].
3. Solve each equation:
- [tex]\( x - 5 = 0 \)[/tex] gives [tex]\( x = 5 \)[/tex].
- [tex]\( x + 5 = 0 \)[/tex] gives [tex]\( x = -5 \)[/tex].
4. For [tex]\( (x^2 + 25) = 0 \)[/tex], solve to find the complex roots: [tex]\( x = \pm 5i \)[/tex].
### Summary of Solutions:
- Problem 10: [tex]\( x = -3, 3, \pm \frac{\sqrt{15}i}{3} \)[/tex]
- Problem 11: [tex]\( x = 0, 2 \)[/tex]
- Problem 12: [tex]\( x = 0, 2, 7 \)[/tex]
- Problem 13: [tex]\( x = -5, 5, \pm 5i \)[/tex]
These are the solutions for each given equation.
