Question

solve for v, where v is a real number.
\sqrt{-11 + 5v} = v - 1
if there is more than one solution, separate them with commas.
if there is no solution, click on
o solution\.
\boxed{v = }

Explanation:

Step1: Square both sides

To eliminate the square root, square both sides of the equation \(\sqrt{-11 + 5v}=v - 1\). We get \((\sqrt{-11 + 5v})^2=(v - 1)^2\), which simplifies to \(-11 + 5v=v^2-2v + 1\).

Step2: Rearrange into quadratic form

Rearrange the equation to standard quadratic form \(ax^2+bx + c = 0\). So, \(v^2-2v + 1+11 - 5v = 0\), which simplifies to \(v^2-7v + 12 = 0\).

Step3: Factor the quadratic

Factor the quadratic equation \(v^2-7v + 12 = 0\). We need two numbers that multiply to \(12\) and add to \(-7\). The numbers are \(-3\) and \(-4\), so the equation factors as \((v - 3)(v - 4)=0\).

Step4: Solve for v

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

Step5: Check for extraneous solutions

• For \(v = 3\): Substitute into the original equation. Left side: \(\sqrt{-11+5\times3}=\sqrt{-11 + 15}=\sqrt{4}=2\). Right side: \(3 - 1 = 2\). So, \(v = 3\) is a valid solution.
• For \(v = 4\): Substitute into the original equation. Left side: \(\sqrt{-11+5\times4}=\sqrt{-11 + 20}=\sqrt{9}=3\). Right side: \(4 - 1 = 3\). So, \(v = 4\) is a valid solution.

Answer:

3, 4