Solve (4y-1)^2=25(1+y)^2

$$( 4 y - 1 ) ^ { 2 } = 25 ( 1 + y ) ^ { 2 }$$

Solve for y

$$y=-\frac{4}{9}\approx -0.444444444$$

$$y=-6$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4y-1\right)^{2}.

$$16y^{2}-8y+1=25\left(1+y\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+y\right)^{2}.

$$16y^{2}-8y+1=25\left(1+2y+y^{2}\right)$$

Use the distributive property to multiply 25 by 1+2y+y^{2}.

$$16y^{2}-8y+1=25+50y+25y^{2}$$

Subtract 25 from both sides.

$$16y^{2}-8y+1-25=50y+25y^{2}$$

Subtract 25 from 1 to get -24.

$$16y^{2}-8y-24=50y+25y^{2}$$

Subtract 50y from both sides.

$$16y^{2}-8y-24-50y=25y^{2}$$

Combine -8y and -50y to get -58y.

$$16y^{2}-58y-24=25y^{2}$$

Subtract 25y^{2} from both sides.

$$16y^{2}-58y-24-25y^{2}=0$$

Combine 16y^{2} and -25y^{2} to get -9y^{2}.

$$-9y^{2}-58y-24=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -9y^{2}+ay+by-24. To find a and b, set up a system to be solved.

$$a+b=-58 ab=-9\left(-24\right)=216$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 216.

$$-1,-216 -2,-108 -3,-72 -4,-54 -6,-36 -8,-27 -9,-24 -12,-18$$

Calculate the sum for each pair.

$$-1-216=-217 -2-108=-110 -3-72=-75 -4-54=-58 -6-36=-42 -8-27=-35 -9-24=-33 -12-18=-30$$

The solution is the pair that gives sum -58.

$$a=-4 b=-54$$

Rewrite -9y^{2}-58y-24 as \left(-9y^{2}-4y\right)+\left(-54y-24\right).

$$\left(-9y^{2}-4y\right)+\left(-54y-24\right)$$

Factor out -y in the first and -6 in the second group.

$$-y\left(9y+4\right)-6\left(9y+4\right)$$

Factor out common term 9y+4 by using distributive property.

$$\left(9y+4\right)\left(-y-6\right)$$

To find equation solutions, solve 9y+4=0 and -y-6=0.

$$y=-\frac{4}{9} y=-6$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4y-1\right)^{2}.

$$16y^{2}-8y+1=25\left(1+y\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+y\right)^{2}.

$$16y^{2}-8y+1=25\left(1+2y+y^{2}\right)$$

Use the distributive property to multiply 25 by 1+2y+y^{2}.

$$16y^{2}-8y+1=25+50y+25y^{2}$$

Subtract 25 from both sides.

$$16y^{2}-8y+1-25=50y+25y^{2}$$

Subtract 25 from 1 to get -24.

$$16y^{2}-8y-24=50y+25y^{2}$$

Subtract 50y from both sides.

$$16y^{2}-8y-24-50y=25y^{2}$$

Combine -8y and -50y to get -58y.

$$16y^{2}-58y-24=25y^{2}$$

Subtract 25y^{2} from both sides.

$$16y^{2}-58y-24-25y^{2}=0$$

Combine 16y^{2} and -25y^{2} to get -9y^{2}.

$$-9y^{2}-58y-24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9 for a, -58 for b, and -24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-58\right)±\sqrt{\left(-58\right)^{2}-4\left(-9\right)\left(-24\right)}}{2\left(-9\right)}$$

Square -58.

$$y=\frac{-\left(-58\right)±\sqrt{3364-4\left(-9\right)\left(-24\right)}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$y=\frac{-\left(-58\right)±\sqrt{3364+36\left(-24\right)}}{2\left(-9\right)}$$

Multiply 36 times -24.

$$y=\frac{-\left(-58\right)±\sqrt{3364-864}}{2\left(-9\right)}$$

Add 3364 to -864.

$$y=\frac{-\left(-58\right)±\sqrt{2500}}{2\left(-9\right)}$$

Take the square root of 2500.

$$y=\frac{-\left(-58\right)±50}{2\left(-9\right)}$$

The opposite of -58 is 58.

$$y=\frac{58±50}{2\left(-9\right)}$$

Multiply 2 times -9.

$$y=\frac{58±50}{-18}$$

Now solve the equation y=\frac{58±50}{-18} when ± is plus. Add 58 to 50.

$$y=\frac{108}{-18}$$

Divide 108 by -18.

$$y=-6$$

Now solve the equation y=\frac{58±50}{-18} when ± is minus. Subtract 50 from 58.

$$y=\frac{8}{-18}$$

Reduce the fraction \frac{8}{-18} to lowest terms by extracting and canceling out 2.

$$y=-\frac{4}{9}$$

The equation is now solved.

$$y=-6 y=-\frac{4}{9}$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4y-1\right)^{2}.

$$16y^{2}-8y+1=25\left(1+y\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+y\right)^{2}.

$$16y^{2}-8y+1=25\left(1+2y+y^{2}\right)$$

Use the distributive property to multiply 25 by 1+2y+y^{2}.

$$16y^{2}-8y+1=25+50y+25y^{2}$$

Subtract 50y from both sides.

$$16y^{2}-8y+1-50y=25+25y^{2}$$

Combine -8y and -50y to get -58y.

$$16y^{2}-58y+1=25+25y^{2}$$

Subtract 25y^{2} from both sides.

$$16y^{2}-58y+1-25y^{2}=25$$

Combine 16y^{2} and -25y^{2} to get -9y^{2}.

$$-9y^{2}-58y+1=25$$

Subtract 1 from both sides.

$$-9y^{2}-58y=25-1$$

Subtract 1 from 25 to get 24.

$$-9y^{2}-58y=24$$

Divide both sides by -9.

$$\frac{-9y^{2}-58y}{-9}=\frac{24}{-9}$$

Dividing by -9 undoes the multiplication by -9.

$$y^{2}+\left(-\frac{58}{-9}\right)y=\frac{24}{-9}$$

Divide -58 by -9.

$$y^{2}+\frac{58}{9}y=\frac{24}{-9}$$

Reduce the fraction \frac{24}{-9} to lowest terms by extracting and canceling out 3.

$$y^{2}+\frac{58}{9}y=-\frac{8}{3}$$

Divide \frac{58}{9}, the coefficient of the x term, by 2 to get \frac{29}{9}. Then add the square of \frac{29}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}+\frac{58}{9}y+\left(\frac{29}{9}\right)^{2}=-\frac{8}{3}+\left(\frac{29}{9}\right)^{2}$$

Square \frac{29}{9} by squaring both the numerator and the denominator of the fraction.

$$y^{2}+\frac{58}{9}y+\frac{841}{81}=-\frac{8}{3}+\frac{841}{81}$$

Add -\frac{8}{3} to \frac{841}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}+\frac{58}{9}y+\frac{841}{81}=\frac{625}{81}$$

Factor y^{2}+\frac{58}{9}y+\frac{841}{81}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y+\frac{29}{9}\right)^{2}=\frac{625}{81}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+\frac{29}{9}\right)^{2}}=\sqrt{\frac{625}{81}}$$

Simplify.

$$y+\frac{29}{9}=\frac{25}{9} y+\frac{29}{9}=-\frac{25}{9}$$

Subtract \frac{29}{9} from both sides of the equation.

$$y=-\frac{4}{9} y=-6$$