Solve (x+4)^2=16(2x-5)^2

$$( x + 4 ) ^ { 2 } = 16 ( 2 x - 5 ) ^ { 2 }$$

Solve for x

$$x = \frac{24}{7} = 3\frac{3}{7} \approx 3.428571429$$

$$x = \frac{16}{9} = 1\frac{7}{9} \approx 1.777777778$$

Steps Using the Quadratic Formula

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

$$x^{2}+8x+16=16\left(2x-5\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2x-5\right)^{2}.

$$x^{2}+8x+16=16\left(4x^{2}-20x+25\right)$$

Use the distributive property to multiply 16 by 4x^{2}-20x+25.

$$x^{2}+8x+16=64x^{2}-320x+400$$

Subtract 64x^{2} from both sides.

$$x^{2}+8x+16-64x^{2}=-320x+400$$

Combine x^{2} and -64x^{2} to get -63x^{2}.

$$-63x^{2}+8x+16=-320x+400$$

Add 320x to both sides.

$$-63x^{2}+8x+16+320x=400$$

Combine 8x and 320x to get 328x.

$$-63x^{2}+328x+16=400$$

Subtract 400 from both sides.

$$-63x^{2}+328x+16-400=0$$

Subtract 400 from 16 to get -384.

$$-63x^{2}+328x-384=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -63 for a, 328 for b, and -384 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-328±\sqrt{328^{2}-4\left(-63\right)\left(-384\right)}}{2\left(-63\right)}$$

Square 328.

$$x=\frac{-328±\sqrt{107584-4\left(-63\right)\left(-384\right)}}{2\left(-63\right)}$$

Multiply -4 times -63.

$$x=\frac{-328±\sqrt{107584+252\left(-384\right)}}{2\left(-63\right)}$$

Multiply 252 times -384.

$$x=\frac{-328±\sqrt{107584-96768}}{2\left(-63\right)}$$

Add 107584 to -96768.

$$x=\frac{-328±\sqrt{10816}}{2\left(-63\right)}$$

Take the square root of 10816.

$$x=\frac{-328±104}{2\left(-63\right)}$$

Multiply 2 times -63.

$$x=\frac{-328±104}{-126}$$

Now solve the equation x=\frac{-328±104}{-126} when ± is plus. Add -328 to 104.

$$x=-\frac{224}{-126}$$

Reduce the fraction \frac{-224}{-126} to lowest terms by extracting and canceling out 14.

$$x=\frac{16}{9}$$

Now solve the equation x=\frac{-328±104}{-126} when ± is minus. Subtract 104 from -328.

$$x=-\frac{432}{-126}$$

Reduce the fraction \frac{-432}{-126} to lowest terms by extracting and canceling out 18.

$$x=\frac{24}{7}$$

The equation is now solved.

$$x=\frac{16}{9} x=\frac{24}{7}$$

Steps for Completing the Square

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

$$x^{2}+8x+16=16\left(2x-5\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2x-5\right)^{2}.

$$x^{2}+8x+16=16\left(4x^{2}-20x+25\right)$$

Use the distributive property to multiply 16 by 4x^{2}-20x+25.

$$x^{2}+8x+16=64x^{2}-320x+400$$

Subtract 64x^{2} from both sides.

$$x^{2}+8x+16-64x^{2}=-320x+400$$

Combine x^{2} and -64x^{2} to get -63x^{2}.

$$-63x^{2}+8x+16=-320x+400$$

Add 320x to both sides.

$$-63x^{2}+8x+16+320x=400$$

Combine 8x and 320x to get 328x.

$$-63x^{2}+328x+16=400$$

Subtract 16 from both sides.

$$-63x^{2}+328x=400-16$$

Subtract 16 from 400 to get 384.

$$-63x^{2}+328x=384$$

Divide both sides by -63.

$$\frac{-63x^{2}+328x}{-63}=\frac{384}{-63}$$

Dividing by -63 undoes the multiplication by -63.

$$x^{2}+\frac{328}{-63}x=\frac{384}{-63}$$

Divide 328 by -63.

$$x^{2}-\frac{328}{63}x=\frac{384}{-63}$$

Reduce the fraction \frac{384}{-63} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{328}{63}x=-\frac{128}{21}$$

Divide -\frac{328}{63}, the coefficient of the x term, by 2 to get -\frac{164}{63}. Then add the square of -\frac{164}{63} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{328}{63}x+\left(-\frac{164}{63}\right)^{2}=-\frac{128}{21}+\left(-\frac{164}{63}\right)^{2}$$

Square -\frac{164}{63} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{328}{63}x+\frac{26896}{3969}=-\frac{128}{21}+\frac{26896}{3969}$$

Add -\frac{128}{21} to \frac{26896}{3969} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{328}{63}x+\frac{26896}{3969}=\frac{2704}{3969}$$

Factor x^{2}-\frac{328}{63}x+\frac{26896}{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}.

$$\left(x-\frac{164}{63}\right)^{2}=\frac{2704}{3969}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{164}{63}\right)^{2}}=\sqrt{\frac{2704}{3969}}$$

Simplify.

$$x-\frac{164}{63}=\frac{52}{63} x-\frac{164}{63}=-\frac{52}{63}$$

Add \frac{164}{63} to both sides of the equation.

$$x=\frac{24}{7} x=\frac{16}{9}$$