Solve left(2x+2right)^2/9+16-4x^2/5=1

$$\frac{ { \left(2x+2 \right) }^{ 2 } }{ 9 } + \frac{ 16-4 { x }^{ 2 } }{ 5 } = 1$$

Solve for x

$$x = -\frac{7}{4} = -1\frac{3}{4} = -1.75$$

$$x = \frac{17}{4} = 4\frac{1}{4} = 4.25$$

Steps Using Factoring By Grouping

Multiply both sides of the equation by 45, the least common multiple of 9,5.

$$5\left(2x+2\right)^{2}+9\left(16-4x^{2}\right)=45$$

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

$$5\left(4x^{2}+8x+4\right)+9\left(16-4x^{2}\right)=45$$

Use the distributive property to multiply 5 by 4x^{2}+8x+4.

$$20x^{2}+40x+20+9\left(16-4x^{2}\right)=45$$

Use the distributive property to multiply 9 by 16-4x^{2}.

$$20x^{2}+40x+20+144-36x^{2}=45$$

Add 20 and 144 to get 164.

$$20x^{2}+40x+164-36x^{2}=45$$

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

$$-16x^{2}+40x+164=45$$

Subtract 45 from both sides.

$$-16x^{2}+40x+164-45=0$$

Subtract 45 from 164 to get 119.

$$-16x^{2}+40x+119=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -16x^{2}+ax+bx+119. To find a and b, set up a system to be solved.

$$a+b=40 ab=-16\times 119=-1904$$

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 -1904.

$$-1,1904 -2,952 -4,476 -7,272 -8,238 -14,136 -16,119 -17,112 -28,68 -34,56$$

Calculate the sum for each pair.

$$-1+1904=1903 -2+952=950 -4+476=472 -7+272=265 -8+238=230 -14+136=122 -16+119=103 -17+112=95 -28+68=40 -34+56=22$$

The solution is the pair that gives sum 40.

$$a=68 b=-28$$

Rewrite -16x^{2}+40x+119 as \left(-16x^{2}+68x\right)+\left(-28x+119\right).

$$\left(-16x^{2}+68x\right)+\left(-28x+119\right)$$

Factor out -4x in the first and -7 in the second group.

$$-4x\left(4x-17\right)-7\left(4x-17\right)$$

Factor out common term 4x-17 by using distributive property.

$$\left(4x-17\right)\left(-4x-7\right)$$

To find equation solutions, solve 4x-17=0 and -4x-7=0.

$$x=\frac{17}{4} x=-\frac{7}{4}$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 45, the least common multiple of 9,5.

$$5\left(2x+2\right)^{2}+9\left(16-4x^{2}\right)=45$$

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

$$5\left(4x^{2}+8x+4\right)+9\left(16-4x^{2}\right)=45$$

Use the distributive property to multiply 5 by 4x^{2}+8x+4.

$$20x^{2}+40x+20+9\left(16-4x^{2}\right)=45$$

Use the distributive property to multiply 9 by 16-4x^{2}.

$$20x^{2}+40x+20+144-36x^{2}=45$$

Add 20 and 144 to get 164.

$$20x^{2}+40x+164-36x^{2}=45$$

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

$$-16x^{2}+40x+164=45$$

Subtract 45 from both sides.

$$-16x^{2}+40x+164-45=0$$

Subtract 45 from 164 to get 119.

$$-16x^{2}+40x+119=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -16 for a, 40 for b, and 119 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-40±\sqrt{40^{2}-4\left(-16\right)\times 119}}{2\left(-16\right)}$$

Square 40.

$$x=\frac{-40±\sqrt{1600-4\left(-16\right)\times 119}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$x=\frac{-40±\sqrt{1600+64\times 119}}{2\left(-16\right)}$$

Multiply 64 times 119.

$$x=\frac{-40±\sqrt{1600+7616}}{2\left(-16\right)}$$

Add 1600 to 7616.

$$x=\frac{-40±\sqrt{9216}}{2\left(-16\right)}$$

Take the square root of 9216.

$$x=\frac{-40±96}{2\left(-16\right)}$$

Multiply 2 times -16.

$$x=\frac{-40±96}{-32}$$

Now solve the equation x=\frac{-40±96}{-32} when ± is plus. Add -40 to 96.

$$x=\frac{56}{-32}$$

Reduce the fraction \frac{56}{-32} to lowest terms by extracting and canceling out 8.

$$x=-\frac{7}{4}$$

Now solve the equation x=\frac{-40±96}{-32} when ± is minus. Subtract 96 from -40.

$$x=-\frac{136}{-32}$$

Reduce the fraction \frac{-136}{-32} to lowest terms by extracting and canceling out 8.

$$x=\frac{17}{4}$$

The equation is now solved.

$$x=-\frac{7}{4} x=\frac{17}{4}$$

Steps for Completing the Square

Multiply both sides of the equation by 45, the least common multiple of 9,5.

$$5\left(2x+2\right)^{2}+9\left(16-4x^{2}\right)=45$$

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

$$5\left(4x^{2}+8x+4\right)+9\left(16-4x^{2}\right)=45$$

Use the distributive property to multiply 5 by 4x^{2}+8x+4.

$$20x^{2}+40x+20+9\left(16-4x^{2}\right)=45$$

Use the distributive property to multiply 9 by 16-4x^{2}.

$$20x^{2}+40x+20+144-36x^{2}=45$$

Add 20 and 144 to get 164.

$$20x^{2}+40x+164-36x^{2}=45$$

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

$$-16x^{2}+40x+164=45$$

Subtract 164 from both sides.

$$-16x^{2}+40x=45-164$$

Subtract 164 from 45 to get -119.

$$-16x^{2}+40x=-119$$

Divide both sides by -16.

$$\frac{-16x^{2}+40x}{-16}=-\frac{119}{-16}$$

Dividing by -16 undoes the multiplication by -16.

$$x^{2}+\frac{40}{-16}x=-\frac{119}{-16}$$

Reduce the fraction \frac{40}{-16} to lowest terms by extracting and canceling out 8.

$$x^{2}-\frac{5}{2}x=-\frac{119}{-16}$$

Divide -119 by -16.

$$x^{2}-\frac{5}{2}x=\frac{119}{16}$$

Divide -\frac{5}{2}, the coefficient of the x term, by 2 to get -\frac{5}{4}. Then add the square of -\frac{5}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{5}{2}x+\left(-\frac{5}{4}\right)^{2}=\frac{119}{16}+\left(-\frac{5}{4}\right)^{2}$$

Square -\frac{5}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{5}{2}x+\frac{25}{16}=\frac{119+25}{16}$$

Add \frac{119}{16} to \frac{25}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{5}{2}x+\frac{25}{16}=9$$

Factor x^{2}-\frac{5}{2}x+\frac{25}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{5}{4}\right)^{2}=9$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{4}\right)^{2}}=\sqrt{9}$$

Simplify.

$$x-\frac{5}{4}=3 x-\frac{5}{4}=-3$$

Add \frac{5}{4} to both sides of the equation.

$$x=\frac{17}{4} x=-\frac{7}{4}$$