Solve 5=19/9-(7x+8)^2

$$5 = \frac { 19 } { 9 } - ( 7 x + 8 ) ^ { 2 }$$

Solve for x (complex solution)

$$x=-\frac{\sqrt{26}i}{21}-\frac{8}{7}\approx -1.142857143-0.242810453i$$

$$x=\frac{\sqrt{26}i}{21}-\frac{8}{7}\approx -1.142857143+0.242810453i$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(7x+8\right)^{2}.

$$5=\frac{19}{9}-\left(49x^{2}+112x+64\right)$$

To find the opposite of 49x^{2}+112x+64, find the opposite of each term.

$$5=\frac{19}{9}-49x^{2}-112x-64$$

Subtract 64 from \frac{19}{9} to get -\frac{557}{9}.

$$5=-\frac{557}{9}-49x^{2}-112x$$

Swap sides so that all variable terms are on the left hand side.

$$-\frac{557}{9}-49x^{2}-112x=5$$

Subtract 5 from both sides.

$$-\frac{557}{9}-49x^{2}-112x-5=0$$

Subtract 5 from -\frac{557}{9} to get -\frac{602}{9}.

$$-\frac{602}{9}-49x^{2}-112x=0$$

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.

$$-49x^{2}-112x-\frac{602}{9}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -49 for a, -112 for b, and -\frac{602}{9} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-112\right)±\sqrt{\left(-112\right)^{2}-4\left(-49\right)\left(-\frac{602}{9}\right)}}{2\left(-49\right)}$$

Square -112.

$$x=\frac{-\left(-112\right)±\sqrt{12544-4\left(-49\right)\left(-\frac{602}{9}\right)}}{2\left(-49\right)}$$

Multiply -4 times -49.

$$x=\frac{-\left(-112\right)±\sqrt{12544+196\left(-\frac{602}{9}\right)}}{2\left(-49\right)}$$

Multiply 196 times -\frac{602}{9}.

$$x=\frac{-\left(-112\right)±\sqrt{12544-\frac{117992}{9}}}{2\left(-49\right)}$$

Add 12544 to -\frac{117992}{9}.

$$x=\frac{-\left(-112\right)±\sqrt{-\frac{5096}{9}}}{2\left(-49\right)}$$

Take the square root of -\frac{5096}{9}.

$$x=\frac{-\left(-112\right)±\frac{14\sqrt{26}i}{3}}{2\left(-49\right)}$$

The opposite of -112 is 112.

$$x=\frac{112±\frac{14\sqrt{26}i}{3}}{2\left(-49\right)}$$

Multiply 2 times -49.

$$x=\frac{112±\frac{14\sqrt{26}i}{3}}{-98}$$

Now solve the equation x=\frac{112±\frac{14\sqrt{26}i}{3}}{-98} when ± is plus. Add 112 to \frac{14i\sqrt{26}}{3}.

$$x=\frac{\frac{14\sqrt{26}i}{3}+112}{-98}$$

Divide 112+\frac{14i\sqrt{26}}{3} by -98.

$$x=-\frac{\sqrt{26}i}{21}-\frac{8}{7}$$

Now solve the equation x=\frac{112±\frac{14\sqrt{26}i}{3}}{-98} when ± is minus. Subtract \frac{14i\sqrt{26}}{3} from 112.

$$x=\frac{-\frac{14\sqrt{26}i}{3}+112}{-98}$$

Divide 112-\frac{14i\sqrt{26}}{3} by -98.

$$x=\frac{\sqrt{26}i}{21}-\frac{8}{7}$$

The equation is now solved.

$$x=-\frac{\sqrt{26}i}{21}-\frac{8}{7} x=\frac{\sqrt{26}i}{21}-\frac{8}{7}$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(7x+8\right)^{2}.

$$5=\frac{19}{9}-\left(49x^{2}+112x+64\right)$$

To find the opposite of 49x^{2}+112x+64, find the opposite of each term.

$$5=\frac{19}{9}-49x^{2}-112x-64$$

Subtract 64 from \frac{19}{9} to get -\frac{557}{9}.

$$5=-\frac{557}{9}-49x^{2}-112x$$

Swap sides so that all variable terms are on the left hand side.

$$-\frac{557}{9}-49x^{2}-112x=5$$

Add \frac{557}{9} to both sides.

$$-49x^{2}-112x=5+\frac{557}{9}$$

Add 5 and \frac{557}{9} to get \frac{602}{9}.

$$-49x^{2}-112x=\frac{602}{9}$$

Divide both sides by -49.

$$\frac{-49x^{2}-112x}{-49}=\frac{\frac{602}{9}}{-49}$$

Dividing by -49 undoes the multiplication by -49.

$$x^{2}+\left(-\frac{112}{-49}\right)x=\frac{\frac{602}{9}}{-49}$$

Reduce the fraction \frac{-112}{-49} to lowest terms by extracting and canceling out 7.

$$x^{2}+\frac{16}{7}x=\frac{\frac{602}{9}}{-49}$$

Divide \frac{602}{9} by -49.

$$x^{2}+\frac{16}{7}x=-\frac{86}{63}$$

Divide \frac{16}{7}, the coefficient of the x term, by 2 to get \frac{8}{7}. Then add the square of \frac{8}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{16}{7}x+\left(\frac{8}{7}\right)^{2}=-\frac{86}{63}+\left(\frac{8}{7}\right)^{2}$$

Square \frac{8}{7} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{16}{7}x+\frac{64}{49}=-\frac{86}{63}+\frac{64}{49}$$

Add -\frac{86}{63} to \frac{64}{49} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{16}{7}x+\frac{64}{49}=-\frac{26}{441}$$

Factor x^{2}+\frac{16}{7}x+\frac{64}{49}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{8}{7}\right)^{2}=-\frac{26}{441}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{8}{7}\right)^{2}}=\sqrt{-\frac{26}{441}}$$

Simplify.

$$x+\frac{8}{7}=\frac{\sqrt{26}i}{21} x+\frac{8}{7}=-\frac{\sqrt{26}i}{21}$$

Subtract \frac{8}{7} from both sides of the equation.

$$x=\frac{\sqrt{26}i}{21}-\frac{8}{7} x=-\frac{\sqrt{26}i}{21}-\frac{8}{7}$$