Solve 4(8y+1)^2-27=(4y+9)(4y-9)+2(5y+2)(2y-7)

$$4 ( 8 y + 1 ) ^ { 2 } - 27 = ( 4 y + 9 ) ( 4 y - 9 ) + 2 ( 5 y + 2 ) ( 2 y - 7 )$$

Solve for y

$$y=\frac{-63+\sqrt{14951}i}{220}\approx -0.286363636+0.55579219i$$

$$y=\frac{-\sqrt{14951}i-63}{220}\approx -0.286363636-0.55579219i$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(8y+1\right)^{2}.

$$4\left(64y^{2}+16y+1\right)-27=\left(4y+9\right)\left(4y-9\right)+2\left(5y+2\right)\left(2y-7\right)$$

Use the distributive property to multiply 4 by 64y^{2}+16y+1.

$$256y^{2}+64y+4-27=\left(4y+9\right)\left(4y-9\right)+2\left(5y+2\right)\left(2y-7\right)$$

Subtract 27 from 4 to get -23.

$$256y^{2}+64y-23=\left(4y+9\right)\left(4y-9\right)+2\left(5y+2\right)\left(2y-7\right)$$

Consider \left(4y+9\right)\left(4y-9\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 9.

$$256y^{2}+64y-23=\left(4y\right)^{2}-81+2\left(5y+2\right)\left(2y-7\right)$$

Expand \left(4y\right)^{2}.

$$256y^{2}+64y-23=4^{2}y^{2}-81+2\left(5y+2\right)\left(2y-7\right)$$

Calculate 4 to the power of 2 and get 16.

$$256y^{2}+64y-23=16y^{2}-81+2\left(5y+2\right)\left(2y-7\right)$$

Use the distributive property to multiply 2 by 5y+2.

$$256y^{2}+64y-23=16y^{2}-81+\left(10y+4\right)\left(2y-7\right)$$

Use the distributive property to multiply 10y+4 by 2y-7 and combine like terms.

$$256y^{2}+64y-23=16y^{2}-81+20y^{2}-62y-28$$

Combine 16y^{2} and 20y^{2} to get 36y^{2}.

$$256y^{2}+64y-23=36y^{2}-81-62y-28$$

Subtract 28 from -81 to get -109.

$$256y^{2}+64y-23=36y^{2}-109-62y$$

Subtract 36y^{2} from both sides.

$$256y^{2}+64y-23-36y^{2}=-109-62y$$

Combine 256y^{2} and -36y^{2} to get 220y^{2}.

$$220y^{2}+64y-23=-109-62y$$

Subtract -109 from both sides.

$$220y^{2}+64y-23-\left(-109\right)=-62y$$

The opposite of -109 is 109.

$$220y^{2}+64y-23+109=-62y$$

Add 62y to both sides.

$$220y^{2}+64y-23+109+62y=0$$

Add -23 and 109 to get 86.

$$220y^{2}+64y+86+62y=0$$

Combine 64y and 62y to get 126y.

$$220y^{2}+126y+86=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 220 for a, 126 for b, and 86 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-126±\sqrt{126^{2}-4\times 220\times 86}}{2\times 220}$$

Square 126.

$$y=\frac{-126±\sqrt{15876-4\times 220\times 86}}{2\times 220}$$

Multiply -4 times 220.

$$y=\frac{-126±\sqrt{15876-880\times 86}}{2\times 220}$$

Multiply -880 times 86.

$$y=\frac{-126±\sqrt{15876-75680}}{2\times 220}$$

Add 15876 to -75680.

$$y=\frac{-126±\sqrt{-59804}}{2\times 220}$$

Take the square root of -59804.

$$y=\frac{-126±2\sqrt{14951}i}{2\times 220}$$

Multiply 2 times 220.

$$y=\frac{-126±2\sqrt{14951}i}{440}$$

Now solve the equation y=\frac{-126±2\sqrt{14951}i}{440} when ± is plus. Add -126 to 2i\sqrt{14951}.

$$y=\frac{-126+2\sqrt{14951}i}{440}$$

Divide -126+2i\sqrt{14951} by 440.

$$y=\frac{-63+\sqrt{14951}i}{220}$$

Now solve the equation y=\frac{-126±2\sqrt{14951}i}{440} when ± is minus. Subtract 2i\sqrt{14951} from -126.

$$y=\frac{-2\sqrt{14951}i-126}{440}$$

Divide -126-2i\sqrt{14951} by 440.

$$y=\frac{-\sqrt{14951}i-63}{220}$$

The equation is now solved.

$$y=\frac{-63+\sqrt{14951}i}{220} y=\frac{-\sqrt{14951}i-63}{220}$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(8y+1\right)^{2}.

$$4\left(64y^{2}+16y+1\right)-27=\left(4y+9\right)\left(4y-9\right)+2\left(5y+2\right)\left(2y-7\right)$$

Use the distributive property to multiply 4 by 64y^{2}+16y+1.

$$256y^{2}+64y+4-27=\left(4y+9\right)\left(4y-9\right)+2\left(5y+2\right)\left(2y-7\right)$$

Subtract 27 from 4 to get -23.

$$256y^{2}+64y-23=\left(4y+9\right)\left(4y-9\right)+2\left(5y+2\right)\left(2y-7\right)$$

Consider \left(4y+9\right)\left(4y-9\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 9.

$$256y^{2}+64y-23=\left(4y\right)^{2}-81+2\left(5y+2\right)\left(2y-7\right)$$

Expand \left(4y\right)^{2}.

$$256y^{2}+64y-23=4^{2}y^{2}-81+2\left(5y+2\right)\left(2y-7\right)$$

Calculate 4 to the power of 2 and get 16.

$$256y^{2}+64y-23=16y^{2}-81+2\left(5y+2\right)\left(2y-7\right)$$

Use the distributive property to multiply 2 by 5y+2.

$$256y^{2}+64y-23=16y^{2}-81+\left(10y+4\right)\left(2y-7\right)$$

Use the distributive property to multiply 10y+4 by 2y-7 and combine like terms.

$$256y^{2}+64y-23=16y^{2}-81+20y^{2}-62y-28$$

Combine 16y^{2} and 20y^{2} to get 36y^{2}.

$$256y^{2}+64y-23=36y^{2}-81-62y-28$$

Subtract 28 from -81 to get -109.

$$256y^{2}+64y-23=36y^{2}-109-62y$$

Subtract 36y^{2} from both sides.

$$256y^{2}+64y-23-36y^{2}=-109-62y$$

Combine 256y^{2} and -36y^{2} to get 220y^{2}.

$$220y^{2}+64y-23=-109-62y$$

Add 62y to both sides.

$$220y^{2}+64y-23+62y=-109$$

Combine 64y and 62y to get 126y.

$$220y^{2}+126y-23=-109$$

Add 23 to both sides.

$$220y^{2}+126y=-109+23$$

Add -109 and 23 to get -86.

$$220y^{2}+126y=-86$$

Divide both sides by 220.

$$\frac{220y^{2}+126y}{220}=-\frac{86}{220}$$

Dividing by 220 undoes the multiplication by 220.

$$y^{2}+\frac{126}{220}y=-\frac{86}{220}$$

Reduce the fraction \frac{126}{220} to lowest terms by extracting and canceling out 2.

$$y^{2}+\frac{63}{110}y=-\frac{86}{220}$$

Reduce the fraction \frac{-86}{220} to lowest terms by extracting and canceling out 2.

$$y^{2}+\frac{63}{110}y=-\frac{43}{110}$$

Divide \frac{63}{110}, the coefficient of the x term, by 2 to get \frac{63}{220}. Then add the square of \frac{63}{220} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}+\frac{63}{110}y+\left(\frac{63}{220}\right)^{2}=-\frac{43}{110}+\left(\frac{63}{220}\right)^{2}$$

Square \frac{63}{220} by squaring both the numerator and the denominator of the fraction.

$$y^{2}+\frac{63}{110}y+\frac{3969}{48400}=-\frac{43}{110}+\frac{3969}{48400}$$

Add -\frac{43}{110} to \frac{3969}{48400} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}+\frac{63}{110}y+\frac{3969}{48400}=-\frac{14951}{48400}$$

Factor y^{2}+\frac{63}{110}y+\frac{3969}{48400}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y+\frac{63}{220}\right)^{2}=-\frac{14951}{48400}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+\frac{63}{220}\right)^{2}}=\sqrt{-\frac{14951}{48400}}$$

Simplify.

$$y+\frac{63}{220}=\frac{\sqrt{14951}i}{220} y+\frac{63}{220}=-\frac{\sqrt{14951}i}{220}$$

Subtract \frac{63}{220} from both sides of the equation.

$$y=\frac{-63+\sqrt{14951}i}{220} y=\frac{-\sqrt{14951}i-63}{220}$$