Question

solve the equation by factoring.\\( v^2 - 18v + 81 = 0 \\)\\( v = \square \\)\\( \text{(use a comma to separate answers as needed.)} \\)

Explanation:

Step1: Identify the quadratic form

The equation \( v^2 - 18v + 81 = 0 \) is a quadratic equation in the form \( ax^2 + bx + c = 0 \), where \( a = 1 \), \( b = -18 \), and \( c = 81 \). We can factor it using the perfect square trinomial formula, which states that \( x^2 - 2xy + y^2=(x - y)^2 \).

Step2: Factor the quadratic

For the given equation, we need to find two numbers that multiply to \( 81 \) (the value of \( c \)) and add up to \( -18 \) (the value of \( b \)). Notice that \( -9 \times -9 = 81 \) and \( -9 + (-9)=-18 \). So, we can factor the quadratic as:
\( v^2 - 18v + 81=(v - 9)(v - 9)=(v - 9)^2 \)

Step3: Solve for \( v \)

Set the factored form equal to zero:
\( (v - 9)^2 = 0 \)
Taking the square root of both sides, we get:
\( v - 9 = 0 \)
Adding \( 9 \) to both sides gives:
\( v = 9 \) (with multiplicity 2, but we just write the solution as \( v = 9 \) since it's a repeated root)

Answer:

9