Solve for $d$ . $\newline$ $\sqrt{9d + 4} = \sqrt{5d}$ $\newline$ $d =$ _____

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Solve radical equations II

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Q. Solve for

d

\newline

\sqrt{9d + 4} = \sqrt{5d}

\newline

d =

_____

Isolate square root terms: First, we need to isolate the square root terms on one side of the equation. We can do this by subtracting $4$ from both sides of the equation. $\newline$ $\sqrt{9d} + 4 - 4 = \sqrt{5d} - 4$ $\newline$ $\sqrt{9d} = \sqrt{5d} - 4$
Square both sides: Next, we square both sides of the equation to eliminate the square roots. $\newline$ $(\sqrt{9d})^2 = (\sqrt{5d} - 4)^2$ $\newline$ $9d = (\sqrt{5d} - 4)(\sqrt{5d} - 4)$
Expand using FOIL method: Now we need to expand the right side of the equation using the FOIL method (First, Outer, Inner, Last). $\newline$ $9d = (\sqrt{5d})(\sqrt{5d}) - 4(\sqrt{5d}) - 4(\sqrt{5d}) + 16$ $\newline$ $9d = 5d - 8\sqrt{5d} + 16$
Isolate square root term: We then isolate the term with the square root on one side of the equation by moving all other terms to the opposite side. $\newline$ $9d - 5d = -8\sqrt{5d} + 16$ $\newline$ $4d = -8\sqrt{5d} + 16$
Move constant term: Next, we move the constant term to the other side by subtracting $16$ from both sides. $\newline$ $4d - 16 = -8\sqrt{5d}$
Divide by $-8$ : Now we divide both sides by $-8$ to isolate $\sqrt{5d}$ . $(4d - 16) / -8 = \sqrt{5d}$ $-0.5d + 2 = \sqrt{5d}$
Square both sides: We square both sides again to eliminate the square root. $\newline$ $(-0.5d + 2)^2 = (\sqrt{5d})^2$ $\newline$ $0.25d^2 - 2d + 4 = 5d$
Set equation to zero: We bring all terms to one side to set the equation to zero and solve for $d$ . $0.25d^2 - 2d + 4 - 5d = 0$ $0.25d^2 - 7d + 4 = 0$
Correct previous step: This is a quadratic equation, and we can solve it by using the quadratic formula, $d = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ , where $a = 0.25$ , $b = -7$ , and $c = 4$ . However, we made a mistake in the previous step. The correct equation after squaring both sides should be: $0.25d^2 - 2d + 4 = 5d$ $0.25d^2 - 7d + 4 = 0$ This is incorrect because the term $-2d$ should have been $-1d$ when we multiplied $-0.5d$ by $-2$ . Let's correct this. $a = 0.25$ $0$ $a = 0.25$ $1$