Sure, let's solve the given equation step-by-step:
Given:
[tex]\[ 2 \sqrt{d - 4} = 6d \][/tex]
Step 1: Isolate the square root term.
To isolate [tex]\(\sqrt{d - 4}\)[/tex], divide both sides of the equation by 2:
[tex]\[ \sqrt{d - 4} = 3d \][/tex]
Step 2: Remove the square root by squaring both sides.
Square both sides of the equation to eliminate the square root:
[tex]\[ (\sqrt{d - 4})^2 = (3d)^2 \][/tex]
[tex]\[ d - 4 = 9d^2 \][/tex]
Step 3: Rearrange the equation into a standard quadratic form.
Move all terms to one side to form a quadratic equation:
[tex]\[ 9d^2 - d - 4 = 0 \][/tex]
Step 4: Identify the coefficients for the quadratic equation.
The quadratic equation is of the form [tex]\(ax^2 + bx + c = 0\)[/tex], where:
[tex]\[ a = 9, \, b = -1, \, c = -4 \][/tex]
Step 5: Solve the quadratic equation using the quadratic formula.
The quadratic formula is:
[tex]\[ d = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Substitute [tex]\(a = 9\)[/tex], [tex]\(b = -1\)[/tex], and [tex]\(c = -4\)[/tex] into the formula:
[tex]\[ d = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 9 \cdot (-4)}}{2 \cdot 9} \][/tex]
[tex]\[ d = \frac{1 \pm \sqrt{1 + 144}}{18} \][/tex]
[tex]\[ d = \frac{1 \pm \sqrt{145}}{18} \][/tex]
Step 6: Calculate the two potential solutions.
[tex]\[ d_1 = \frac{1 + \sqrt{145}}{18} \approx 0.7245330321551275 \][/tex]
[tex]\[ d_2 = \frac{1 - \sqrt{145}}{18} \approx -0.6134219210440164 \][/tex]
To summarize, the solutions to the equation [tex]\(2 \sqrt{d - 4} = 6d\)[/tex] are:
[tex]\[ d \approx 0.7245330321551275 \][/tex]
and
[tex]\[ d \approx -0.6134219210440164 \][/tex]
