Solve (x+3)(x-3)-29(2x-29)=0

$$( x + 3 ) ( x - 3 ) - 29 ( 2 x - 29 ) = 0$$

Solve for x

$$x=26$$

$$x=32$$

Steps Using Factoring

Consider \left(x+3\right)\left(x-3\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 3.

$$x^{2}-9-29\left(2x-29\right)=0$$

Use the distributive property to multiply -29 by 2x-29.

$$x^{2}-9-58x+841=0$$

Add -9 and 841 to get 832.

$$x^{2}+832-58x=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$x^{2}-58x+832=0$$

To solve the equation, factor x^{2}-58x+832 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=-58 ab=832$$

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 832.

$$-1,-832 -2,-416 -4,-208 -8,-104 -13,-64 -16,-52 -26,-32$$

Calculate the sum for each pair.

$$-1-832=-833 -2-416=-418 -4-208=-212 -8-104=-112 -13-64=-77 -16-52=-68 -26-32=-58$$

The solution is the pair that gives sum -58.

$$a=-32 b=-26$$

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

$$\left(x-32\right)\left(x-26\right)$$

To find equation solutions, solve x-32=0 and x-26=0.

$$x=32 x=26$$

Steps Using Factoring By Grouping

Consider \left(x+3\right)\left(x-3\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 3.

$$x^{2}-9-29\left(2x-29\right)=0$$

Use the distributive property to multiply -29 by 2x-29.

$$x^{2}-9-58x+841=0$$

Add -9 and 841 to get 832.

$$x^{2}+832-58x=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$x^{2}-58x+832=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+832. To find a and b, set up a system to be solved.

$$a+b=-58 ab=1\times 832=832$$

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 832.

$$-1,-832 -2,-416 -4,-208 -8,-104 -13,-64 -16,-52 -26,-32$$

Calculate the sum for each pair.

$$-1-832=-833 -2-416=-418 -4-208=-212 -8-104=-112 -13-64=-77 -16-52=-68 -26-32=-58$$

The solution is the pair that gives sum -58.

$$a=-32 b=-26$$

Rewrite x^{2}-58x+832 as \left(x^{2}-32x\right)+\left(-26x+832\right).

$$\left(x^{2}-32x\right)+\left(-26x+832\right)$$

Factor out x in the first and -26 in the second group.

$$x\left(x-32\right)-26\left(x-32\right)$$

Factor out common term x-32 by using distributive property.

$$\left(x-32\right)\left(x-26\right)$$

To find equation solutions, solve x-32=0 and x-26=0.

$$x=32 x=26$$

Steps Using the Quadratic Formula

Consider \left(x+3\right)\left(x-3\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 3.

$$x^{2}-9-29\left(2x-29\right)=0$$

Use the distributive property to multiply -29 by 2x-29.

$$x^{2}-9-58x+841=0$$

Add -9 and 841 to get 832.

$$x^{2}+832-58x=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x^{2}-58x+832=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -58 for b, and 832 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-58\right)±\sqrt{\left(-58\right)^{2}-4\times 832}}{2}$$

Square -58.

$$x=\frac{-\left(-58\right)±\sqrt{3364-4\times 832}}{2}$$

Multiply -4 times 832.

$$x=\frac{-\left(-58\right)±\sqrt{3364-3328}}{2}$$

Add 3364 to -3328.

$$x=\frac{-\left(-58\right)±\sqrt{36}}{2}$$

Take the square root of 36.

$$x=\frac{-\left(-58\right)±6}{2}$$

The opposite of -58 is 58.

$$x=\frac{58±6}{2}$$

Now solve the equation x=\frac{58±6}{2} when ± is plus. Add 58 to 6.

$$x=\frac{64}{2}$$

Divide 64 by 2.

$$x=32$$

Now solve the equation x=\frac{58±6}{2} when ± is minus. Subtract 6 from 58.

$$x=\frac{52}{2}$$

Divide 52 by 2.

$$x=26$$

The equation is now solved.

$$x=32 x=26$$

Steps for Completing the Square

Consider \left(x+3\right)\left(x-3\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 3.

$$x^{2}-9-29\left(2x-29\right)=0$$

Use the distributive property to multiply -29 by 2x-29.

$$x^{2}-9-58x+841=0$$

Add -9 and 841 to get 832.

$$x^{2}+832-58x=0$$

Subtract 832 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}-58x=-832$$

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

$$x^{2}-58x+\left(-29\right)^{2}=-832+\left(-29\right)^{2}$$

Square -29.

$$x^{2}-58x+841=-832+841$$

Add -832 to 841.

$$x^{2}-58x+841=9$$

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

$$\left(x-29\right)^{2}=9$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-29\right)^{2}}=\sqrt{9}$$

Simplify.

$$x-29=3 x-29=-3$$

Add 29 to both sides of the equation.

$$x=32 x=26$$