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Answer :
Sure! Let's solve each of the equations step-by-step:
10. Solve [tex]\(3x^2 - 27 = 0\)[/tex]
1. Add 27 to both sides to isolate the [tex]\(x^2\)[/tex] term:
[tex]\[ 3x^2 = 27 \][/tex]
2. Divide both sides by 3:
[tex]\[ x^2 = 9 \][/tex]
3. Take the square root of both sides:
[tex]\[ x = \pm 3 \][/tex]
Solutions are [tex]\(x = 3\)[/tex] and [tex]\(x = -3\)[/tex].
11. Solve [tex]\(5x^2 - 10x = 0\)[/tex]
1. Factor out the greatest common factor, which is 5x:
[tex]\[ 5x(x - 2) = 0 \][/tex]
2. Set each factor equal to zero:
- [tex]\(5x = 0\)[/tex] leads to [tex]\(x = 0\)[/tex]
- [tex]\(x - 2 = 0\)[/tex] leads to [tex]\(x = 2\)[/tex]
Solutions are [tex]\(x = 0\)[/tex] and [tex]\(x = 2\)[/tex].
12. Solve [tex]\(3x^2 = 2x\)[/tex]
1. Move [tex]\(2x\)[/tex] to the left side:
[tex]\[ 3x^2 - 2x = 0 \][/tex]
2. Factor out the greatest common factor, which is x:
[tex]\[ x(3x - 2) = 0 \][/tex]
3. Set each factor equal to zero:
- [tex]\(x = 0\)[/tex]
- [tex]\(3x - 2 = 0\)[/tex] gives [tex]\(x = \frac{2}{3}\)[/tex]
Solutions are [tex]\(x = 0\)[/tex] and [tex]\(x = \frac{2}{3}\)[/tex].
13. Solve [tex]\(-2x^2 = x\)[/tex]
1. Move [tex]\(x\)[/tex] to the left side:
[tex]\[ -2x^2 - x = 0 \][/tex]
2. Factor out the greatest common factor, which is [tex]\(-x\)[/tex]:
[tex]\[ -x(2x + 1) = 0 \][/tex]
3. Set each factor equal to zero:
- [tex]\(-x = 0\)[/tex] leads to [tex]\(x = 0\)[/tex]
- [tex]\(2x + 1 = 0\)[/tex] gives [tex]\(x = -\frac{1}{2}\)[/tex]
Solutions are [tex]\(x = 0\)[/tex] and [tex]\(x = -\frac{1}{2}\)[/tex].
14. Solve [tex]\(x^2 + 11 = 27\)[/tex]
1. Subtract 11 from both sides:
[tex]\[ x^2 = 16 \][/tex]
2. Take the square root of both sides:
[tex]\[ x = \pm 4 \][/tex]
Solutions are [tex]\(x = 4\)[/tex] and [tex]\(x = -4\)[/tex].
15. Solve [tex]\(3x^2 + 7x = 2x^2\)[/tex]
1. Move [tex]\(2x^2\)[/tex] to the left side:
[tex]\[ 3x^2 + 7x - 2x^2 = 0 \][/tex]
2. Simplify the equation:
[tex]\[ x^2 + 7x = 0 \][/tex]
3. Factor the equation:
[tex]\[ x(x + 7) = 0 \][/tex]
4. Set each factor equal to zero:
- [tex]\(x = 0\)[/tex]
- [tex]\(x + 7 = 0\)[/tex] gives [tex]\(x = -7\)[/tex]
Solutions are [tex]\(x = 0\)[/tex] and [tex]\(x = -7\)[/tex].
16. Solve [tex]\(6x^2 + 4x - 8 = x^2 + 4x + 1\)[/tex]
1. Move all terms to one side:
[tex]\[ 6x^2 + 4x - 8 - x^2 - 4x - 1 = 0 \][/tex]
2. Simplify the equation:
[tex]\[ 5x^2 - 9 = 0 \][/tex]
3. Add 9 to both sides:
[tex]\[ 5x^2 = 9 \][/tex]
4. Divide by 5:
[tex]\[ x^2 = \frac{9}{5} \][/tex]
5. Take the square root of both sides:
[tex]\[ x = \pm \sqrt{\frac{9}{5}} = \pm \frac{3\sqrt{5}}{5} \][/tex]
Solutions are [tex]\(x = \frac{3\sqrt{5}}{5}\)[/tex] and [tex]\(x = -\frac{3\sqrt{5}}{5}\)[/tex].
10. Solve [tex]\(3x^2 - 27 = 0\)[/tex]
1. Add 27 to both sides to isolate the [tex]\(x^2\)[/tex] term:
[tex]\[ 3x^2 = 27 \][/tex]
2. Divide both sides by 3:
[tex]\[ x^2 = 9 \][/tex]
3. Take the square root of both sides:
[tex]\[ x = \pm 3 \][/tex]
Solutions are [tex]\(x = 3\)[/tex] and [tex]\(x = -3\)[/tex].
11. Solve [tex]\(5x^2 - 10x = 0\)[/tex]
1. Factor out the greatest common factor, which is 5x:
[tex]\[ 5x(x - 2) = 0 \][/tex]
2. Set each factor equal to zero:
- [tex]\(5x = 0\)[/tex] leads to [tex]\(x = 0\)[/tex]
- [tex]\(x - 2 = 0\)[/tex] leads to [tex]\(x = 2\)[/tex]
Solutions are [tex]\(x = 0\)[/tex] and [tex]\(x = 2\)[/tex].
12. Solve [tex]\(3x^2 = 2x\)[/tex]
1. Move [tex]\(2x\)[/tex] to the left side:
[tex]\[ 3x^2 - 2x = 0 \][/tex]
2. Factor out the greatest common factor, which is x:
[tex]\[ x(3x - 2) = 0 \][/tex]
3. Set each factor equal to zero:
- [tex]\(x = 0\)[/tex]
- [tex]\(3x - 2 = 0\)[/tex] gives [tex]\(x = \frac{2}{3}\)[/tex]
Solutions are [tex]\(x = 0\)[/tex] and [tex]\(x = \frac{2}{3}\)[/tex].
13. Solve [tex]\(-2x^2 = x\)[/tex]
1. Move [tex]\(x\)[/tex] to the left side:
[tex]\[ -2x^2 - x = 0 \][/tex]
2. Factor out the greatest common factor, which is [tex]\(-x\)[/tex]:
[tex]\[ -x(2x + 1) = 0 \][/tex]
3. Set each factor equal to zero:
- [tex]\(-x = 0\)[/tex] leads to [tex]\(x = 0\)[/tex]
- [tex]\(2x + 1 = 0\)[/tex] gives [tex]\(x = -\frac{1}{2}\)[/tex]
Solutions are [tex]\(x = 0\)[/tex] and [tex]\(x = -\frac{1}{2}\)[/tex].
14. Solve [tex]\(x^2 + 11 = 27\)[/tex]
1. Subtract 11 from both sides:
[tex]\[ x^2 = 16 \][/tex]
2. Take the square root of both sides:
[tex]\[ x = \pm 4 \][/tex]
Solutions are [tex]\(x = 4\)[/tex] and [tex]\(x = -4\)[/tex].
15. Solve [tex]\(3x^2 + 7x = 2x^2\)[/tex]
1. Move [tex]\(2x^2\)[/tex] to the left side:
[tex]\[ 3x^2 + 7x - 2x^2 = 0 \][/tex]
2. Simplify the equation:
[tex]\[ x^2 + 7x = 0 \][/tex]
3. Factor the equation:
[tex]\[ x(x + 7) = 0 \][/tex]
4. Set each factor equal to zero:
- [tex]\(x = 0\)[/tex]
- [tex]\(x + 7 = 0\)[/tex] gives [tex]\(x = -7\)[/tex]
Solutions are [tex]\(x = 0\)[/tex] and [tex]\(x = -7\)[/tex].
16. Solve [tex]\(6x^2 + 4x - 8 = x^2 + 4x + 1\)[/tex]
1. Move all terms to one side:
[tex]\[ 6x^2 + 4x - 8 - x^2 - 4x - 1 = 0 \][/tex]
2. Simplify the equation:
[tex]\[ 5x^2 - 9 = 0 \][/tex]
3. Add 9 to both sides:
[tex]\[ 5x^2 = 9 \][/tex]
4. Divide by 5:
[tex]\[ x^2 = \frac{9}{5} \][/tex]
5. Take the square root of both sides:
[tex]\[ x = \pm \sqrt{\frac{9}{5}} = \pm \frac{3\sqrt{5}}{5} \][/tex]
Solutions are [tex]\(x = \frac{3\sqrt{5}}{5}\)[/tex] and [tex]\(x = -\frac{3\sqrt{5}}{5}\)[/tex].
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