Solve y-5=sqrt{-6y+46}

$$y - 5 = \sqrt { - 6 y + 46 }$$

Solve for y

$$y=7$$

Solution Steps

Square both sides of the equation.

$$\left(y-5\right)^{2}=\left(\sqrt{-6y+46}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(y-5\right)^{2}.

$$y^{2}-10y+25=\left(\sqrt{-6y+46}\right)^{2}$$

Calculate \sqrt{-6y+46} to the power of 2 and get -6y+46.

$$y^{2}-10y+25=-6y+46$$

Add 6y to both sides.

$$y^{2}-10y+25+6y=46$$

Combine -10y and 6y to get -4y.

$$y^{2}-4y+25=46$$

Subtract 46 from both sides.

$$y^{2}-4y+25-46=0$$

Subtract 46 from 25 to get -21.

$$y^{2}-4y-21=0$$

To solve the equation, factor y^{2}-4y-21 using formula y^{2}+\left(a+b\right)y+ab=\left(y+a\right)\left(y+b\right). To find a and b, set up a system to be solved.

$$a+b=-4 ab=-21$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -21.

$$1,-21 3,-7$$

Calculate the sum for each pair.

$$1-21=-20 3-7=-4$$

The solution is the pair that gives sum -4.

$$a=-7 b=3$$

Rewrite factored expression \left(y+a\right)\left(y+b\right) using the obtained values.

$$\left(y-7\right)\left(y+3\right)$$

To find equation solutions, solve y-7=0 and y+3=0.

$$y=7 y=-3$$

Substitute 7 for y in the equation y-5=\sqrt{-6y+46}.

$$7-5=\sqrt{-6\times 7+46}$$

Simplify. The value y=7 satisfies the equation.

$$2=2$$

Substitute -3 for y in the equation y-5=\sqrt{-6y+46}.

$$-3-5=\sqrt{-6\left(-3\right)+46}$$

Simplify. The value y=-3 does not satisfy the equation because the left and the right hand side have opposite signs.

$$-8=8$$

Equation y-5=\sqrt{46-6y} has a unique solution.

$$y=7$$