Solve by Factoring 128y = square root of 1-192y

$128 y = \sqrt{1 - 192 y}$

Step 1

Subtract $\sqrt{1 - 192 y}$ from both sides of the equation.

$128 y - \sqrt{1 - 192 y} = 0$

Step 2

Subtract $128 y$ from both sides of the equation.

$- \sqrt{1 - 192 y} = - 128 y$

Step 3

To remove the radical on the left side of the equation, square both sides of the equation.

${(- \sqrt{1 - 192 y})}^{2} = {(- 128 y)}^{2}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1 - 192 y}$ as ${(1 - 192 y)}^{\frac{1}{2}}$ .

${(- {(1 - 192 y)}^{\frac{1}{2}})}^{2} = {(- 128 y)}^{2}$

Simplify the left side.

Simplify ${(- {(1 - 192 y)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- {(1 - 192 y)}^{\frac{1}{2}}$ .

${(- 1)}^{2} {({(1 - 192 y)}^{\frac{1}{2}})}^{2} = {(- 128 y)}^{2}$

Raise $- 1$ to the power of $2$ .

$1 {({(1 - 192 y)}^{\frac{1}{2}})}^{2} = {(- 128 y)}^{2}$

Multiply ${({(1 - 192 y)}^{\frac{1}{2}})}^{2}$ by $1$ .

${({(1 - 192 y)}^{\frac{1}{2}})}^{2} = {(- 128 y)}^{2}$

Multiply the exponents in ${({(1 - 192 y)}^{\frac{1}{2}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(1 - 192 y)}^{\frac{1}{2} \cdot 2} = {(- 128 y)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(1 - 192 y)}^{\frac{1}{2} \cdot 2} = {(- 128 y)}^{2}$

Rewrite the expression.

${(1 - 192 y)}^{1} = {(- 128 y)}^{2}$

Simplify.

$1 - 192 y = {(- 128 y)}^{2}$

Simplify the right side.

Simplify ${(- 128 y)}^{2}$ .

Apply the product rule to $- 128 y$ .

$1 - 192 y = {(- 128)}^{2} y^{2}$

Raise $- 128$ to the power of $2$ .

$1 - 192 y = 16384 y^{2}$

Step 5

Solve for $y$ .

Subtract $16384 y^{2}$ from both sides of the equation.

$1 - 192 y - 16384 y^{2} = 0$

Factor the left side of the equation.

Factor $- 1$ out of $1 - 192 y - 16384 y^{2}$ .

Reorder the expression.

Move $1$ .

$- 192 y - 16384 y^{2} + 1 = 0$

Reorder $- 192 y$ and $- 16384 y^{2}$ .

$- 16384 y^{2} - 192 y + 1 = 0$

Factor $- 1$ out of $- 16384 y^{2}$ .

$- (16384 y^{2}) - 192 y + 1 = 0$

Factor $- 1$ out of $- 192 y$ .

$- (16384 y^{2}) - (192 y) + 1 = 0$

Rewrite $1$ as $- 1 (- 1)$ .

$- (16384 y^{2}) - (192 y) - 1 \cdot - 1 = 0$

Factor $- 1$ out of $- (16384 y^{2}) - (192 y)$ .

$- (16384 y^{2} + 192 y) - 1 \cdot - 1 = 0$

Factor $- 1$ out of $- (16384 y^{2} + 192 y) - 1 (- 1)$ .

$- (16384 y^{2} + 192 y - 1) = 0$

Factor.

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 16384 \cdot - 1 = - 16384$ and whose sum is $b = 192$ .

Factor $192$ out of $192 y$ .

$- (16384 y^{2} + 192 (y) - 1) = 0$

Rewrite $192$ as $- 64$ plus $256$

$- (16384 y^{2} + (- 64 + 256) y - 1) = 0$

Apply the distributive property.

$- (16384 y^{2} - 64 y + 256 y - 1) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$- ((16384 y^{2} - 64 y) + 256 y - 1) = 0$

Factor out the greatest common factor (GCF) from each group.

$- (64 y (256 y - 1) + 1 (256 y - 1)) = 0$

Factor the polynomial by factoring out the greatest common factor, $256 y - 1$ .

$- ((256 y - 1) (64 y + 1)) = 0$

Remove unnecessary parentheses.

$- (256 y - 1) (64 y + 1) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$256 y - 1 = 0$

$64 y + 1 = 0$

Set $256 y - 1$ equal to $0$ and solve for $y$ .

Set $256 y - 1$ equal to $0$ .

$256 y - 1 = 0$

Solve $256 y - 1 = 0$ for $y$ .

Add $1$ to both sides of the equation.

$256 y = 1$

Divide each term in $256 y = 1$ by $256$ and simplify.

Divide each term in $256 y = 1$ by $256$ .

$\frac{256 y}{256} = \frac{1}{256}$

Simplify the left side.

Cancel the common factor of $256$ .

Cancel the common factor.

$\frac{256 y}{256} = \frac{1}{256}$

Divide $y$ by $1$ .

$y = \frac{1}{256}$

Set $64 y + 1$ equal to $0$ and solve for $y$ .

Set $64 y + 1$ equal to $0$ .

$64 y + 1 = 0$

Solve $64 y + 1 = 0$ for $y$ .

Subtract $1$ from both sides of the equation.

$64 y = - 1$

Divide each term in $64 y = - 1$ by $64$ and simplify.

Divide each term in $64 y = - 1$ by $64$ .

$\frac{64 y}{64} = \frac{- 1}{64}$

Simplify the left side.

Cancel the common factor of $64$ .

Cancel the common factor.

$\frac{64 y}{64} = \frac{- 1}{64}$

Divide $y$ by $1$ .

$y = \frac{- 1}{64}$

Simplify the right side.

Move the negative in front of the fraction.

$y = - \frac{1}{64}$

The final solution is all the values that make $- (256 y - 1) (64 y + 1) = 0$ true.

$y = \frac{1}{256}, - \frac{1}{64}$

Step 6

Exclude the solutions that do not make $128 y - \sqrt{1 - 192 y} = 0$ true.

$y = \frac{1}{256}$

Solve by Factoring 128y = square root of 1-192y

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