6. Solve each of these equations and check each solution: $$ \begin{array}{ll} ext { (i) } \sqrt{x+5}=5-\sqrt{x} & ext { (ii) } \sqrt{5 x+6}=\sqrt{2 x}+2 \ ext { (iii) } \sqrt{x+7}+\sqrt{x}=7 & ext { (iv) } \sqrt{3 x-2}=\sqrt{x-2}+2\end{array} $$

Question

6. Solve each of these equations and check each solution: $$ \begin{array}{ll}	ext { (i) } \sqrt{x+5}=5-\sqrt{x} & 	ext { (ii) } \sqrt{5 x+6}=\sqrt{2 x}+2 \ 	ext { (iii) } \sqrt{x+7}+\sqrt{x}=7 & 	ext { (iv) } \sqrt{3 x-2}=\sqrt{x-2}+2\end{array} $$

UpStudy AI · Accepted Answer

Solve the equation $$ \sqrt{x+5}=5-\sqrt{x} $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$\sqrt{x+5}=5-\sqrt{x}$$
- step1: Find the domain:
  $$\sqrt{x+5}=5-\sqrt{x},x\geq 0$$
- step2: Evaluate:
  $$\sqrt{x+5}=5-\sqrt{x},5-\sqrt{x}\geq 0$$
- step3: Evaluate:
  $$\sqrt{x+5}=5-\sqrt{x},x\leq 25$$
- step4: Solve the equation:
  $$x=4,x\leq 25$$
- step5: Find the intersection:
  $$x=4$$
- step6: Check if the solution is in the defined range:
  $$x=4,x\geq 0$$
- step7: Find the intersection:
  $$x=4$$
- step8: Check the solution:
  $$x=4$$
Solve the equation $$ \sqrt{5x+6}=\sqrt{2x}+2 $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$\sqrt{5x+6}=\sqrt{2x}+2$$
- step1: Find the domain:
  $$\sqrt{5x+6}=\sqrt{2x}+2,x\geq 0$$
- step2: Raise both sides to the $$2$$-th power$$:$$
  $$\left(\sqrt{5x+6}ight)^{2}=\left(\sqrt{2x}+2ight)^{2}$$
- step3: Evaluate the power:
  $$5x+6=2x+4\sqrt{2x}+4$$
- step4: Move the expression to the left side:
  $$5x-2x=4\sqrt{2x}+4-6$$
- step5: Calculate:
  $$5x-2x=4\sqrt{2x}-2$$
- step6: Calculate:
  $$3x=4\sqrt{2x}-2$$
- step7: Swap the sides:
  $$4\sqrt{2x}-2=3x$$
- step8: Move the constant to the right-hand side:
  $$4\sqrt{2x}=3x+2$$
- step9: Rewrite the expression:
  $$\sqrt{2x}=\frac{3x+2}{4}$$
- step10: Evaluate:
  $$\sqrt{2x}=\frac{3x+2}{4},\frac{3x+2}{4}\geq 0$$
- step11: Evaluate:
  $$\sqrt{2x}=\frac{3x+2}{4},x\geq -\frac{2}{3}$$
- step12: Solve the equation:
  $$\begin{align}&x=2\&x=\frac{2}{9}\end{align},x\geq -\frac{2}{3}$$
- step13: Find the intersection:
  $$\begin{align}&x=2\&x=\frac{2}{9}\end{align}$$
- step14: Check if the solution is in the defined range:
  $$\begin{align}&x=2\&x=\frac{2}{9}\end{align},x\geq 0$$
- step15: Find the intersection:
  $$\begin{align}&x=2\&x=\frac{2}{9}\end{align}$$
- step16: Check the solution:
  $$\begin{align}&x=2\&x=\frac{2}{9}\end{align}$$
- step17: Rewrite:
  $$x_{1}=\frac{2}{9},x_{2}=2$$
Solve the equation $$ \sqrt{x+7}+\sqrt{x}=7 $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$\sqrt{x+7}+\sqrt{x}=7$$
- step1: Find the domain:
  $$\sqrt{x+7}+\sqrt{x}=7,x\geq 0$$
- step2: Move the expression to the right-hand side:
  $$\sqrt{x+7}=7-\sqrt{x}$$
- step3: Evaluate:
  $$\sqrt{x+7}=7-\sqrt{x},7-\sqrt{x}\geq 0$$
- step4: Evaluate:
  $$\sqrt{x+7}=7-\sqrt{x},x\leq 49$$
- step5: Solve the equation:
  $$x=9,x\leq 49$$
- step6: Find the intersection:
  $$x=9$$
- step7: Check if the solution is in the defined range:
  $$x=9,x\geq 0$$
- step8: Find the intersection:
  $$x=9$$
- step9: Check the solution:
  $$x=9$$
Solve the equation $$ \sqrt{3x-2}=\sqrt{x-2}+2 $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$\sqrt{3x-2}=\sqrt{x-2}+2$$
- step1: Find the domain:
  $$\sqrt{3x-2}=\sqrt{x-2}+2,x\geq 2$$
- step2: Raise both sides to the $$2$$-th power$$:$$
  $$\left(\sqrt{3x-2}ight)^{2}=\left(\sqrt{x-2}+2ight)^{2}$$
- step3: Evaluate the power:
  $$3x-2=x+2+4\sqrt{x-2}$$
- step4: Move the expression to the left side:
  $$3x-x=2+4\sqrt{x-2}-\left(-2ight)$$
- step5: Calculate:
  $$3x-x=4+4\sqrt{x-2}$$
- step6: Calculate:
  $$2x=4+4\sqrt{x-2}$$
- step7: Swap the sides:
  $$4+4\sqrt{x-2}=2x$$
- step8: Move the constant to the right-hand side:
  $$4\sqrt{x-2}=2x-4$$
- step9: Divide both sides:
  $$2\sqrt{x-2}=x-2$$
- step10: Rewrite the expression:
  $$\sqrt{x-2}=\frac{x-2}{2}$$
- step11: Evaluate:
  $$\sqrt{x-2}=\frac{x-2}{2},\frac{x-2}{2}\geq 0$$
- step12: Evaluate:
  $$\sqrt{x-2}=\frac{x-2}{2},x\geq 2$$
- step13: Solve the equation:
  $$\begin{align}&x=6\&x=2\end{align},x\geq 2$$
- step14: Find the intersection:
  $$\begin{align}&x=6\&x=2\end{align}$$
- step15: Check the solution:
  $$\begin{align}&x=6\&x=2\end{align}$$
- step16: Rewrite:
  $$x_{1}=2,x_{2}=6$$
The solutions to the given equations are as follows:

(i) $$ x = 4 $$

(ii) $$ x = \frac{2}{9} $$ or $$ x = 2 $$

(iii) $$ x = 9 $$

(iv) $$ x = 2 $$ or $$ x = 6 $$
(i) $$ x = 4 $$ (ii) $$ x = \frac{2}{9} $$ or $$ x = 2 $$ (iii) $$ x = 9 $$ (iv) $$ x = 2 $$ or $$ x = 6 $$

Solve each of these equations and check each solution:

Upstudy AI Solution

Answer

Solution

Bonus Knowledge

Related Questions

Latest Algebra Questions