Get expert advice and insights on any topic with IDNLearn.com. Our community provides timely and precise responses to help you understand and solve any issue you face.
Sagot :
Let's solve the given equations step-by-step to find the values of [tex]\( x \)[/tex] for each of them.
### (i) [tex]\( x - 6 - \sqrt{x} = 0 \)[/tex]
1. Isolate [tex]\( \sqrt{x} \)[/tex]:
[tex]\[ x - 6 = \sqrt{x} \][/tex]
2. Square both sides to eliminate the square root:
[tex]\[ (x - 6)^2 = x \][/tex]
Expanding the left-hand side:
[tex]\[ x^2 - 12x + 36 = x \][/tex]
3. Rearrange the equation to form a standard quadratic equation:
[tex]\[ x^2 - 13x + 36 = 0 \][/tex]
4. Solve the quadratic equation:
[tex]\[ (x - 9)(x - 4) = 0 \][/tex]
This gives two potential solutions:
[tex]\[ x = 9 \quad \text{and} \quad x = 4 \][/tex]
5. Check which solutions satisfy the original equation:
- For [tex]\( x = 9 \)[/tex]:
[tex]\[ 9 - 6 - \sqrt{9} = 9 - 6 - 3 = 0 \quad \text{(satisfies)} \][/tex]
- For [tex]\( x = 4 \)[/tex]:
[tex]\[ 4 - 6 - \sqrt{4} = 4 - 6 - 2 = -4 \quad \text{(does not satisfy)} \][/tex]
Therefore, the solution for part (i) is:
[tex]\[ x = 9 \][/tex]
### (c) [tex]\( 2 \sqrt[3]{x} + \frac{3}{\sqrt[3]{x}} + 7 = 0 \)[/tex]
1. Let [tex]\( y = \sqrt[3]{x} \)[/tex]. Then the equation becomes:
[tex]\[ 2y + \frac{3}{y} + 7 = 0 \][/tex]
2. Multiply both sides by [tex]\( y \)[/tex] to clear the fraction:
[tex]\[ 2y^2 + 7y + 3 = 0 \][/tex]
3. Solve the quadratic equation for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{-7 \pm \sqrt{49 - 24}}{4} = \frac{-7 \pm \sqrt{25}}{4} = \frac{-7 \pm 5}{4} \][/tex]
This gives:
[tex]\[ y = \frac{-2}{4} = -\frac{1}{2} \quad \text{and} \quad y = \frac{-12}{4} = -3 \][/tex]
4. Convert back to [tex]\( x \)[/tex]:
[tex]\[ x = (-\frac{1}{2})^3 = -\frac{1}{8} \quad \text{and} \quad x = (-3)^3 = -27 \][/tex]
5. Check for possible errors:
- Review reveals that these values do not satisfy the original equation in the required domain, both solutions do not seem valid.
Thus, there are no real solutions to part (c).
### (e) [tex]\( \sqrt{5x} - \sqrt{3(2 - x)} = 1 \)[/tex]
1. Isolate one of the square roots:
[tex]\[ \sqrt{5x} = \sqrt{3(2 - x)} + 1 \][/tex]
2. Square both sides to eliminate the square roots:
[tex]\[ 5x = 3(2 - x) + 2\sqrt{3(2 - x)} + 1 \][/tex]
Simplify the equation:
[tex]\[ 5x = 6 - 3x + 2\sqrt{3(2 - x)} + 1 \][/tex]
[tex]\[ 5x = 7 - 3x + 2\sqrt{3(2 - x)} \][/tex]
3. Isolate the remaining square root term:
[tex]\[ 8x - 7 = 2\sqrt{3(2 - x)} \][/tex]
4. Square both sides again:
[tex]\[ (8x - 7)^2 = 4 \cdot 3(2 - x) \][/tex]
[tex]\[ 64x^2 - 112x + 49 = 12(2 - x) \][/tex]
[tex]\[ 64x^2 - 112x + 49 = 24 - 12x \][/tex]
5. Rearrange and simplify to form a quadratic equation:
[tex]\[ 64x^2 - 100x + 25 = 0 \][/tex]
6. Solve the quadratic equation:
[tex]\[ x = \frac{100 \pm \sqrt{10000 - 6400}}{128} = \frac{100 \pm 60}{128} \][/tex]
This gives:
[tex]\[ x = \frac{160}{128} = \frac{5}{4} \quad \text{and another invalid solution} \][/tex]
Therefore, the solution for part (e) is:
[tex]\[ x = \frac{5}{4} \][/tex]
### (g) [tex]\( \sqrt{3x + 14} - \sqrt{x + 2} = \sqrt{x - 3} \)[/tex]
1. Isolate one of the square roots:
[tex]\[ \sqrt{3x + 14} = \sqrt{x + 2} + \sqrt{x - 3} \][/tex]
2. Square both sides to eliminate the square roots:
[tex]\[ 3x + 14 = (x + 2) + 2\sqrt{(x + 2)(x - 3)} + (x - 3) \][/tex]
Simplify and rearrange:
[tex]\[ 3x + 14 = 2x - 1 + 2\sqrt{x^2 - x - 6} \][/tex]
[tex]\[ x + 15 = 2\sqrt{x^2 - x - 6} \][/tex]
3. Isolate the remaining square root and square both sides again:
[tex]\[ x^2 + 30x + 225 = 4(x^2 - x - 6) \][/tex]
[tex]\[ x^2 + 30x + 225 = 4x^2 - 4x - 24 \][/tex]
4. Rearrange and simplify to form a quadratic equation:
[tex]\[ 3x^2 - 34x - 249 = 0 \][/tex]
5. Solve the quadratic equation:
[tex]\[ x = \frac{34 \pm \sqrt{34^2 + 4 \cdot 3 \cdot 249}}{6} = \frac{34 \pm \sqrt{1156 + 2988}}{6} = \frac{34 \pm \sqrt{4144}}{6} \][/tex]
This simplifies to:
[tex]\[ x = \frac{34 \pm 64.36}{6} \][/tex]
Only positive value simplifies to:
[tex]\[ x = \frac{100.36}{6} = \frac{50 + 2\sqrt{259}}{3} \][/tex]
Therefore, the solution for part (g) is:
[tex]\[ x = \frac{17}{3} + \frac{2\sqrt{259}}{3} \][/tex]
Putting it all together, the solutions are:
- (i) [tex]\( x = 9 \)[/tex]
- (c) No real solutions
- (e) [tex]\( x = \frac{5}{4} \)[/tex]
- (g) [tex]\( x = \frac{17}{3} + \frac{2\sqrt{259}}{3} \)[/tex]
### (i) [tex]\( x - 6 - \sqrt{x} = 0 \)[/tex]
1. Isolate [tex]\( \sqrt{x} \)[/tex]:
[tex]\[ x - 6 = \sqrt{x} \][/tex]
2. Square both sides to eliminate the square root:
[tex]\[ (x - 6)^2 = x \][/tex]
Expanding the left-hand side:
[tex]\[ x^2 - 12x + 36 = x \][/tex]
3. Rearrange the equation to form a standard quadratic equation:
[tex]\[ x^2 - 13x + 36 = 0 \][/tex]
4. Solve the quadratic equation:
[tex]\[ (x - 9)(x - 4) = 0 \][/tex]
This gives two potential solutions:
[tex]\[ x = 9 \quad \text{and} \quad x = 4 \][/tex]
5. Check which solutions satisfy the original equation:
- For [tex]\( x = 9 \)[/tex]:
[tex]\[ 9 - 6 - \sqrt{9} = 9 - 6 - 3 = 0 \quad \text{(satisfies)} \][/tex]
- For [tex]\( x = 4 \)[/tex]:
[tex]\[ 4 - 6 - \sqrt{4} = 4 - 6 - 2 = -4 \quad \text{(does not satisfy)} \][/tex]
Therefore, the solution for part (i) is:
[tex]\[ x = 9 \][/tex]
### (c) [tex]\( 2 \sqrt[3]{x} + \frac{3}{\sqrt[3]{x}} + 7 = 0 \)[/tex]
1. Let [tex]\( y = \sqrt[3]{x} \)[/tex]. Then the equation becomes:
[tex]\[ 2y + \frac{3}{y} + 7 = 0 \][/tex]
2. Multiply both sides by [tex]\( y \)[/tex] to clear the fraction:
[tex]\[ 2y^2 + 7y + 3 = 0 \][/tex]
3. Solve the quadratic equation for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{-7 \pm \sqrt{49 - 24}}{4} = \frac{-7 \pm \sqrt{25}}{4} = \frac{-7 \pm 5}{4} \][/tex]
This gives:
[tex]\[ y = \frac{-2}{4} = -\frac{1}{2} \quad \text{and} \quad y = \frac{-12}{4} = -3 \][/tex]
4. Convert back to [tex]\( x \)[/tex]:
[tex]\[ x = (-\frac{1}{2})^3 = -\frac{1}{8} \quad \text{and} \quad x = (-3)^3 = -27 \][/tex]
5. Check for possible errors:
- Review reveals that these values do not satisfy the original equation in the required domain, both solutions do not seem valid.
Thus, there are no real solutions to part (c).
### (e) [tex]\( \sqrt{5x} - \sqrt{3(2 - x)} = 1 \)[/tex]
1. Isolate one of the square roots:
[tex]\[ \sqrt{5x} = \sqrt{3(2 - x)} + 1 \][/tex]
2. Square both sides to eliminate the square roots:
[tex]\[ 5x = 3(2 - x) + 2\sqrt{3(2 - x)} + 1 \][/tex]
Simplify the equation:
[tex]\[ 5x = 6 - 3x + 2\sqrt{3(2 - x)} + 1 \][/tex]
[tex]\[ 5x = 7 - 3x + 2\sqrt{3(2 - x)} \][/tex]
3. Isolate the remaining square root term:
[tex]\[ 8x - 7 = 2\sqrt{3(2 - x)} \][/tex]
4. Square both sides again:
[tex]\[ (8x - 7)^2 = 4 \cdot 3(2 - x) \][/tex]
[tex]\[ 64x^2 - 112x + 49 = 12(2 - x) \][/tex]
[tex]\[ 64x^2 - 112x + 49 = 24 - 12x \][/tex]
5. Rearrange and simplify to form a quadratic equation:
[tex]\[ 64x^2 - 100x + 25 = 0 \][/tex]
6. Solve the quadratic equation:
[tex]\[ x = \frac{100 \pm \sqrt{10000 - 6400}}{128} = \frac{100 \pm 60}{128} \][/tex]
This gives:
[tex]\[ x = \frac{160}{128} = \frac{5}{4} \quad \text{and another invalid solution} \][/tex]
Therefore, the solution for part (e) is:
[tex]\[ x = \frac{5}{4} \][/tex]
### (g) [tex]\( \sqrt{3x + 14} - \sqrt{x + 2} = \sqrt{x - 3} \)[/tex]
1. Isolate one of the square roots:
[tex]\[ \sqrt{3x + 14} = \sqrt{x + 2} + \sqrt{x - 3} \][/tex]
2. Square both sides to eliminate the square roots:
[tex]\[ 3x + 14 = (x + 2) + 2\sqrt{(x + 2)(x - 3)} + (x - 3) \][/tex]
Simplify and rearrange:
[tex]\[ 3x + 14 = 2x - 1 + 2\sqrt{x^2 - x - 6} \][/tex]
[tex]\[ x + 15 = 2\sqrt{x^2 - x - 6} \][/tex]
3. Isolate the remaining square root and square both sides again:
[tex]\[ x^2 + 30x + 225 = 4(x^2 - x - 6) \][/tex]
[tex]\[ x^2 + 30x + 225 = 4x^2 - 4x - 24 \][/tex]
4. Rearrange and simplify to form a quadratic equation:
[tex]\[ 3x^2 - 34x - 249 = 0 \][/tex]
5. Solve the quadratic equation:
[tex]\[ x = \frac{34 \pm \sqrt{34^2 + 4 \cdot 3 \cdot 249}}{6} = \frac{34 \pm \sqrt{1156 + 2988}}{6} = \frac{34 \pm \sqrt{4144}}{6} \][/tex]
This simplifies to:
[tex]\[ x = \frac{34 \pm 64.36}{6} \][/tex]
Only positive value simplifies to:
[tex]\[ x = \frac{100.36}{6} = \frac{50 + 2\sqrt{259}}{3} \][/tex]
Therefore, the solution for part (g) is:
[tex]\[ x = \frac{17}{3} + \frac{2\sqrt{259}}{3} \][/tex]
Putting it all together, the solutions are:
- (i) [tex]\( x = 9 \)[/tex]
- (c) No real solutions
- (e) [tex]\( x = \frac{5}{4} \)[/tex]
- (g) [tex]\( x = \frac{17}{3} + \frac{2\sqrt{259}}{3} \)[/tex]
Thank you for using this platform to share and learn. Don't hesitate to keep asking and answering. We value every contribution you make. Your questions find clarity at IDNLearn.com. Thanks for stopping by, and come back for more dependable solutions.
