Question

solve for x: √(12 - x) = x
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x =
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Explanation:

Step1: Square both sides

To eliminate the square root, square both sides of the equation \(\sqrt{12 - x}=x\). We get \((\sqrt{12 - x})^2 = x^2\), which simplifies to \(12 - x = x^2\).

Step2: Rearrange into quadratic equation

Rearrange the equation \(12 - x = x^2\) to standard quadratic form \(ax^2+bx + c = 0\). So, \(x^2+x - 12=0\).

Step3: Factor the quadratic equation

Factor the quadratic equation \(x^2+x - 12 = 0\). We need two numbers that multiply to \(- 12\) and add to \(1\). The numbers are \(4\) and \(-3\). So, the equation factors as \((x + 4)(x - 3)=0\).

Step4: Solve for x

Set each factor equal to zero: \(x+4 = 0\) or \(x - 3=0\). This gives \(x=-4\) or \(x = 3\).

Step5: Check for extraneous solutions

We need to check these solutions in the original equation because squaring both sides can introduce extraneous solutions.

• For \(x=-4\): Substitute into \(\sqrt{12 - x}=x\). Left - hand side: \(\sqrt{12-(-4)}=\sqrt{16} = 4\). Right - hand side: \(-4\). Since \(4

eq - 4\), \(x = - 4\) is extraneous.

• For \(x = 3\): Substitute into \(\sqrt{12 - x}=x\). Left - hand side: \(\sqrt{12 - 3}=\sqrt{9}=3\). Right - hand side: \(3\). Since \(3 = 3\), \(x = 3\) is a valid solution.

Answer:

\(3\)