Question

solve the radical equation. check all proposed solutions.\\(\sqrt{x + 7} = x - 5\\)

Explanation:

Step1: Square both sides to eliminate radical

To solve the radical equation \(\sqrt{x + 7}=x - 5\), we first square both sides of the equation. Squaring the left side \((\sqrt{x + 7})^2\) gives \(x + 7\), and squaring the right side \((x - 5)^2\) using the formula \((a - b)^2=a^2-2ab + b^2\) (where \(a = x\) and \(b = 5\)) gives \(x^2-10x + 25\). So we have the equation:
\[x + 7=x^2-10x + 25\]

Step2: Rearrange into quadratic equation

Subtract \(x\) and \(7\) from both sides to set the equation to zero:
\[x^2-10x + 25-x - 7 = 0\]
Simplify the left side:
\[x^2-11x + 18 = 0\]

Step3: Factor the quadratic equation

We need to find two numbers that multiply to \(18\) and add up to \(- 11\). The numbers are \(-2\) and \(-9\) because \((-2)\times(-9)=18\) and \(-2+(-9)=-11\). So we can factor the quadratic as:
\[(x - 2)(x - 9)=0\]

Step4: Solve for x

Using the zero - product property, if \(ab = 0\), then either \(a = 0\) or \(b = 0\). So we set each factor equal to zero:

• For \(x - 2=0\), we get \(x = 2\).
• For \(x - 9=0\), we get \(x = 9\).

Step5: Check the solutions

Check \(x = 2\):

Substitute \(x = 2\) into the original equation \(\sqrt{x + 7}=x - 5\).
Left side: \(\sqrt{2+7}=\sqrt{9}=3\).
Right side: \(2 - 5=-3\).
Since \(3
eq - 3\), \(x = 2\) is an extraneous solution.

Check \(x = 9\):

Substitute \(x = 9\) into the original equation.
Left side: \(\sqrt{9 + 7}=\sqrt{16}=4\).
Right side: \(9 - 5 = 4\).
Since the left side equals the right side, \(x = 9\) is a valid solution.

Answer:

\(x = 9\)