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

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

Solve for x

$$x = \frac{7}{2} = 3\frac{1}{2} = 3.5$$

$$x=\frac{3}{8}=0.375$$

Steps Using Factoring By Grouping

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

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

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

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

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

$$9x^{2}+12x+4=25x^{2}-50x+25$$

Subtract 25x^{2} from both sides.

$$9x^{2}+12x+4-25x^{2}=-50x+25$$

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

$$-16x^{2}+12x+4=-50x+25$$

Add 50x to both sides.

$$-16x^{2}+12x+4+50x=25$$

Combine 12x and 50x to get 62x.

$$-16x^{2}+62x+4=25$$

Subtract 25 from both sides.

$$-16x^{2}+62x+4-25=0$$

Subtract 25 from 4 to get -21.

$$-16x^{2}+62x-21=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -16x^{2}+ax+bx-21. To find a and b, set up a system to be solved.

$$a+b=62 ab=-16\left(-21\right)=336$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 336.

$$1,336 2,168 3,112 4,84 6,56 7,48 8,42 12,28 14,24 16,21$$

Calculate the sum for each pair.

$$1+336=337 2+168=170 3+112=115 4+84=88 6+56=62 7+48=55 8+42=50 12+28=40 14+24=38 16+21=37$$

The solution is the pair that gives sum 62.

$$a=56 b=6$$

Rewrite -16x^{2}+62x-21 as \left(-16x^{2}+56x\right)+\left(6x-21\right).

$$\left(-16x^{2}+56x\right)+\left(6x-21\right)$$

Factor out -8x in the first and 3 in the second group.

$$-8x\left(2x-7\right)+3\left(2x-7\right)$$

Factor out common term 2x-7 by using distributive property.

$$\left(2x-7\right)\left(-8x+3\right)$$

To find equation solutions, solve 2x-7=0 and -8x+3=0.

$$x=\frac{7}{2} x=\frac{3}{8}$$

Steps Using the Quadratic Formula

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

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

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

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

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

$$9x^{2}+12x+4=25x^{2}-50x+25$$

Subtract 25x^{2} from both sides.

$$9x^{2}+12x+4-25x^{2}=-50x+25$$

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

$$-16x^{2}+12x+4=-50x+25$$

Add 50x to both sides.

$$-16x^{2}+12x+4+50x=25$$

Combine 12x and 50x to get 62x.

$$-16x^{2}+62x+4=25$$

Subtract 25 from both sides.

$$-16x^{2}+62x+4-25=0$$

Subtract 25 from 4 to get -21.

$$-16x^{2}+62x-21=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -16 for a, 62 for b, and -21 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-62±\sqrt{62^{2}-4\left(-16\right)\left(-21\right)}}{2\left(-16\right)}$$

Square 62.

$$x=\frac{-62±\sqrt{3844-4\left(-16\right)\left(-21\right)}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$x=\frac{-62±\sqrt{3844+64\left(-21\right)}}{2\left(-16\right)}$$

Multiply 64 times -21.

$$x=\frac{-62±\sqrt{3844-1344}}{2\left(-16\right)}$$

Add 3844 to -1344.

$$x=\frac{-62±\sqrt{2500}}{2\left(-16\right)}$$

Take the square root of 2500.

$$x=\frac{-62±50}{2\left(-16\right)}$$

Multiply 2 times -16.

$$x=\frac{-62±50}{-32}$$

Now solve the equation x=\frac{-62±50}{-32} when ± is plus. Add -62 to 50.

$$x=-\frac{12}{-32}$$

Reduce the fraction \frac{-12}{-32} to lowest terms by extracting and canceling out 4.

$$x=\frac{3}{8}$$

Now solve the equation x=\frac{-62±50}{-32} when ± is minus. Subtract 50 from -62.

$$x=-\frac{112}{-32}$$

Reduce the fraction \frac{-112}{-32} to lowest terms by extracting and canceling out 16.

$$x=\frac{7}{2}$$

The equation is now solved.

$$x=\frac{3}{8} x=\frac{7}{2}$$

Steps for Completing the Square

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

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

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

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

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

$$9x^{2}+12x+4=25x^{2}-50x+25$$

Subtract 25x^{2} from both sides.

$$9x^{2}+12x+4-25x^{2}=-50x+25$$

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

$$-16x^{2}+12x+4=-50x+25$$

Add 50x to both sides.

$$-16x^{2}+12x+4+50x=25$$

Combine 12x and 50x to get 62x.

$$-16x^{2}+62x+4=25$$

Subtract 4 from both sides.

$$-16x^{2}+62x=25-4$$

Subtract 4 from 25 to get 21.

$$-16x^{2}+62x=21$$

Divide both sides by -16.

$$\frac{-16x^{2}+62x}{-16}=\frac{21}{-16}$$

Dividing by -16 undoes the multiplication by -16.

$$x^{2}+\frac{62}{-16}x=\frac{21}{-16}$$

Reduce the fraction \frac{62}{-16} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{31}{8}x=\frac{21}{-16}$$

Divide 21 by -16.

$$x^{2}-\frac{31}{8}x=-\frac{21}{16}$$

Divide -\frac{31}{8}, the coefficient of the x term, by 2 to get -\frac{31}{16}. Then add the square of -\frac{31}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{31}{8}x+\left(-\frac{31}{16}\right)^{2}=-\frac{21}{16}+\left(-\frac{31}{16}\right)^{2}$$

Square -\frac{31}{16} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{31}{8}x+\frac{961}{256}=-\frac{21}{16}+\frac{961}{256}$$

Add -\frac{21}{16} to \frac{961}{256} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{31}{8}x+\frac{961}{256}=\frac{625}{256}$$

Factor x^{2}-\frac{31}{8}x+\frac{961}{256}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{31}{16}\right)^{2}=\frac{625}{256}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{31}{16}\right)^{2}}=\sqrt{\frac{625}{256}}$$

Simplify.

$$x-\frac{31}{16}=\frac{25}{16} x-\frac{31}{16}=-\frac{25}{16}$$

Add \frac{31}{16} to both sides of the equation.

$$x=\frac{7}{2} x=\frac{3}{8}$$