Solve 4(x+1)^2-169=0

$$4 ( x + 1 ) ^ { 2 } - 169 = 0$$

Solve for x

$$x = \frac{11}{2} = 5\frac{1}{2} = 5.5$$

$$x = -\frac{15}{2} = -7\frac{1}{2} = -7.5$$

Steps Using Factoring By Grouping

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

$$4\left(x^{2}+2x+1\right)-169=0$$

Use the distributive property to multiply 4 by x^{2}+2x+1.

$$4x^{2}+8x+4-169=0$$

Subtract 169 from 4 to get -165.

$$4x^{2}+8x-165=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 4x^{2}+ax+bx-165. To find a and b, set up a system to be solved.

$$a+b=8 ab=4\left(-165\right)=-660$$

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 -660.

$$-1,660 -2,330 -3,220 -4,165 -5,132 -6,110 -10,66 -11,60 -12,55 -15,44 -20,33 -22,30$$

Calculate the sum for each pair.

$$-1+660=659 -2+330=328 -3+220=217 -4+165=161 -5+132=127 -6+110=104 -10+66=56 -11+60=49 -12+55=43 -15+44=29 -20+33=13 -22+30=8$$

The solution is the pair that gives sum 8.

$$a=-22 b=30$$

Rewrite 4x^{2}+8x-165 as \left(4x^{2}-22x\right)+\left(30x-165\right).

$$\left(4x^{2}-22x\right)+\left(30x-165\right)$$

Factor out 2x in the first and 15 in the second group.

$$2x\left(2x-11\right)+15\left(2x-11\right)$$

Factor out common term 2x-11 by using distributive property.

$$\left(2x-11\right)\left(2x+15\right)$$

To find equation solutions, solve 2x-11=0 and 2x+15=0.

$$x=\frac{11}{2} x=-\frac{15}{2}$$

Steps Using the Quadratic Formula

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

$$4\left(x^{2}+2x+1\right)-169=0$$

Use the distributive property to multiply 4 by x^{2}+2x+1.

$$4x^{2}+8x+4-169=0$$

Subtract 169 from 4 to get -165.

$$4x^{2}+8x-165=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 8 for b, and -165 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\times 4\left(-165\right)}}{2\times 4}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\times 4\left(-165\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-8±\sqrt{64-16\left(-165\right)}}{2\times 4}$$

Multiply -16 times -165.

$$x=\frac{-8±\sqrt{64+2640}}{2\times 4}$$

Add 64 to 2640.

$$x=\frac{-8±\sqrt{2704}}{2\times 4}$$

Take the square root of 2704.

$$x=\frac{-8±52}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-8±52}{8}$$

Now solve the equation x=\frac{-8±52}{8} when ± is plus. Add -8 to 52.

$$x=\frac{44}{8}$$

Reduce the fraction \frac{44}{8} to lowest terms by extracting and canceling out 4.

$$x=\frac{11}{2}$$

Now solve the equation x=\frac{-8±52}{8} when ± is minus. Subtract 52 from -8.

$$x=-\frac{60}{8}$$

Reduce the fraction \frac{-60}{8} to lowest terms by extracting and canceling out 4.

$$x=-\frac{15}{2}$$

The equation is now solved.

$$x=\frac{11}{2} x=-\frac{15}{2}$$

Steps for Completing the Square

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

$$4\left(x^{2}+2x+1\right)-169=0$$

Use the distributive property to multiply 4 by x^{2}+2x+1.

$$4x^{2}+8x+4-169=0$$

Subtract 169 from 4 to get -165.

$$4x^{2}+8x-165=0$$

Add 165 to both sides. Anything plus zero gives itself.

$$4x^{2}+8x=165$$

Divide both sides by 4.

$$\frac{4x^{2}+8x}{4}=\frac{165}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\frac{8}{4}x=\frac{165}{4}$$

Divide 8 by 4.

$$x^{2}+2x=\frac{165}{4}$$

Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+2x+1^{2}=\frac{165}{4}+1^{2}$$

Square 1.

$$x^{2}+2x+1=\frac{165}{4}+1$$

Add \frac{165}{4} to 1.

$$x^{2}+2x+1=\frac{169}{4}$$

Factor x^{2}+2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+1\right)^{2}=\frac{169}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{\frac{169}{4}}$$

Simplify.

$$x+1=\frac{13}{2} x+1=-\frac{13}{2}$$

Subtract 1 from both sides of the equation.

$$x=\frac{11}{2} x=-\frac{15}{2}$$