Solve 9(2x+3)^2=25(1-3x)^2

$$9 ( 2 x + 3 ) ^ { 2 } = 25 ( 1 - 3 x ) ^ { 2 }$$

Solve for x

$$x = \frac{14}{9} = 1\frac{5}{9} \approx 1.555555556$$

$$x=-\frac{4}{21}\approx -0.19047619$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+3\right)^{2}.

$$9\left(4x^{2}+12x+9\right)=25\left(1-3x\right)^{2}$$

Use the distributive property to multiply 9 by 4x^{2}+12x+9.

$$36x^{2}+108x+81=25\left(1-3x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1-3x\right)^{2}.

$$36x^{2}+108x+81=25\left(1-6x+9x^{2}\right)$$

Use the distributive property to multiply 25 by 1-6x+9x^{2}.

$$36x^{2}+108x+81=25-150x+225x^{2}$$

Subtract 25 from both sides.

$$36x^{2}+108x+81-25=-150x+225x^{2}$$

Subtract 25 from 81 to get 56.

$$36x^{2}+108x+56=-150x+225x^{2}$$

Add 150x to both sides.

$$36x^{2}+108x+56+150x=225x^{2}$$

Combine 108x and 150x to get 258x.

$$36x^{2}+258x+56=225x^{2}$$

Subtract 225x^{2} from both sides.

$$36x^{2}+258x+56-225x^{2}=0$$

Combine 36x^{2} and -225x^{2} to get -189x^{2}.

$$-189x^{2}+258x+56=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -189x^{2}+ax+bx+56. To find a and b, set up a system to be solved.

$$a+b=258 ab=-189\times 56=-10584$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -10584.

$$-1,10584 -2,5292 -3,3528 -4,2646 -6,1764 -7,1512 -8,1323 -9,1176 -12,882 -14,756 -18,588 -21,504 -24,441 -27,392 -28,378 -36,294 -42,252 -49,216 -54,196 -56,189 -63,168 -72,147 -84,126 -98,108$$

Calculate the sum for each pair.

$$-1+10584=10583 -2+5292=5290 -3+3528=3525 -4+2646=2642 -6+1764=1758 -7+1512=1505 -8+1323=1315 -9+1176=1167 -12+882=870 -14+756=742 -18+588=570 -21+504=483 -24+441=417 -27+392=365 -28+378=350 -36+294=258 -42+252=210 -49+216=167 -54+196=142 -56+189=133 -63+168=105 -72+147=75 -84+126=42 -98+108=10$$

The solution is the pair that gives sum 258.

$$a=294 b=-36$$

Rewrite -189x^{2}+258x+56 as \left(-189x^{2}+294x\right)+\left(-36x+56\right).

$$\left(-189x^{2}+294x\right)+\left(-36x+56\right)$$

Factor out -21x in the first and -4 in the second group.

$$-21x\left(9x-14\right)-4\left(9x-14\right)$$

Factor out common term 9x-14 by using distributive property.

$$\left(9x-14\right)\left(-21x-4\right)$$

To find equation solutions, solve 9x-14=0 and -21x-4=0.

$$x=\frac{14}{9} x=-\frac{4}{21}$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+3\right)^{2}.

$$9\left(4x^{2}+12x+9\right)=25\left(1-3x\right)^{2}$$

Use the distributive property to multiply 9 by 4x^{2}+12x+9.

$$36x^{2}+108x+81=25\left(1-3x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1-3x\right)^{2}.

$$36x^{2}+108x+81=25\left(1-6x+9x^{2}\right)$$

Use the distributive property to multiply 25 by 1-6x+9x^{2}.

$$36x^{2}+108x+81=25-150x+225x^{2}$$

Subtract 25 from both sides.

$$36x^{2}+108x+81-25=-150x+225x^{2}$$

Subtract 25 from 81 to get 56.

$$36x^{2}+108x+56=-150x+225x^{2}$$

Add 150x to both sides.

$$36x^{2}+108x+56+150x=225x^{2}$$

Combine 108x and 150x to get 258x.

$$36x^{2}+258x+56=225x^{2}$$

Subtract 225x^{2} from both sides.

$$36x^{2}+258x+56-225x^{2}=0$$

Combine 36x^{2} and -225x^{2} to get -189x^{2}.

$$-189x^{2}+258x+56=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -189 for a, 258 for b, and 56 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-258±\sqrt{258^{2}-4\left(-189\right)\times 56}}{2\left(-189\right)}$$

Square 258.

$$x=\frac{-258±\sqrt{66564-4\left(-189\right)\times 56}}{2\left(-189\right)}$$

Multiply -4 times -189.

$$x=\frac{-258±\sqrt{66564+756\times 56}}{2\left(-189\right)}$$

Multiply 756 times 56.

$$x=\frac{-258±\sqrt{66564+42336}}{2\left(-189\right)}$$

Add 66564 to 42336.

$$x=\frac{-258±\sqrt{108900}}{2\left(-189\right)}$$

Take the square root of 108900.

$$x=\frac{-258±330}{2\left(-189\right)}$$

Multiply 2 times -189.

$$x=\frac{-258±330}{-378}$$

Now solve the equation x=\frac{-258±330}{-378} when ± is plus. Add -258 to 330.

$$x=\frac{72}{-378}$$

Reduce the fraction \frac{72}{-378} to lowest terms by extracting and canceling out 18.

$$x=-\frac{4}{21}$$

Now solve the equation x=\frac{-258±330}{-378} when ± is minus. Subtract 330 from -258.

$$x=-\frac{588}{-378}$$

Reduce the fraction \frac{-588}{-378} to lowest terms by extracting and canceling out 42.

$$x=\frac{14}{9}$$

The equation is now solved.

$$x=-\frac{4}{21} x=\frac{14}{9}$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+3\right)^{2}.

$$9\left(4x^{2}+12x+9\right)=25\left(1-3x\right)^{2}$$

Use the distributive property to multiply 9 by 4x^{2}+12x+9.

$$36x^{2}+108x+81=25\left(1-3x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1-3x\right)^{2}.

$$36x^{2}+108x+81=25\left(1-6x+9x^{2}\right)$$

Use the distributive property to multiply 25 by 1-6x+9x^{2}.

$$36x^{2}+108x+81=25-150x+225x^{2}$$

Add 150x to both sides.

$$36x^{2}+108x+81+150x=25+225x^{2}$$

Combine 108x and 150x to get 258x.

$$36x^{2}+258x+81=25+225x^{2}$$

Subtract 225x^{2} from both sides.

$$36x^{2}+258x+81-225x^{2}=25$$

Combine 36x^{2} and -225x^{2} to get -189x^{2}.

$$-189x^{2}+258x+81=25$$

Subtract 81 from both sides.

$$-189x^{2}+258x=25-81$$

Subtract 81 from 25 to get -56.

$$-189x^{2}+258x=-56$$

Divide both sides by -189.

$$\frac{-189x^{2}+258x}{-189}=-\frac{56}{-189}$$

Dividing by -189 undoes the multiplication by -189.

$$x^{2}+\frac{258}{-189}x=-\frac{56}{-189}$$

Reduce the fraction \frac{258}{-189} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{86}{63}x=-\frac{56}{-189}$$

Reduce the fraction \frac{-56}{-189} to lowest terms by extracting and canceling out 7.

$$x^{2}-\frac{86}{63}x=\frac{8}{27}$$

Divide -\frac{86}{63}, the coefficient of the x term, by 2 to get -\frac{43}{63}. Then add the square of -\frac{43}{63} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{86}{63}x+\left(-\frac{43}{63}\right)^{2}=\frac{8}{27}+\left(-\frac{43}{63}\right)^{2}$$

Square -\frac{43}{63} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{86}{63}x+\frac{1849}{3969}=\frac{8}{27}+\frac{1849}{3969}$$

Add \frac{8}{27} to \frac{1849}{3969} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{86}{63}x+\frac{1849}{3969}=\frac{3025}{3969}$$

Factor x^{2}-\frac{86}{63}x+\frac{1849}{3969}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{43}{63}\right)^{2}=\frac{3025}{3969}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{43}{63}\right)^{2}}=\sqrt{\frac{3025}{3969}}$$

Simplify.

$$x-\frac{43}{63}=\frac{55}{63} x-\frac{43}{63}=-\frac{55}{63}$$

Add \frac{43}{63} to both sides of the equation.

$$x=\frac{14}{9} x=-\frac{4}{21}$$