Solve 96x^2-(14x+3)^2+80x=-5

$$96 x ^ { 2 } - ( 14 x + 3 ) ^ { 2 } + 80 x = - 5$$

Solve for x (complex solution)

$$x=\frac{-1+3\sqrt{11}i}{50}\approx -0.02+0.198997487i$$

$$x=\frac{-3\sqrt{11}i-1}{50}\approx -0.02-0.198997487i$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(14x+3\right)^{2}.

$$96x^{2}-\left(196x^{2}+84x+9\right)+80x=-5$$

To find the opposite of 196x^{2}+84x+9, find the opposite of each term.

$$96x^{2}-196x^{2}-84x-9+80x=-5$$

Combine 96x^{2} and -196x^{2} to get -100x^{2}.

$$-100x^{2}-84x-9+80x=-5$$

Combine -84x and 80x to get -4x.

$$-100x^{2}-4x-9=-5$$

Add 5 to both sides.

$$-100x^{2}-4x-9+5=0$$

Add -9 and 5 to get -4.

$$-100x^{2}-4x-4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -100 for a, -4 for b, and -4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-100\right)\left(-4\right)}}{2\left(-100\right)}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-100\right)\left(-4\right)}}{2\left(-100\right)}$$

Multiply -4 times -100.

$$x=\frac{-\left(-4\right)±\sqrt{16+400\left(-4\right)}}{2\left(-100\right)}$$

Multiply 400 times -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-1600}}{2\left(-100\right)}$$

Add 16 to -1600.

$$x=\frac{-\left(-4\right)±\sqrt{-1584}}{2\left(-100\right)}$$

Take the square root of -1584.

$$x=\frac{-\left(-4\right)±12\sqrt{11}i}{2\left(-100\right)}$$

The opposite of -4 is 4.

$$x=\frac{4±12\sqrt{11}i}{2\left(-100\right)}$$

Multiply 2 times -100.

$$x=\frac{4±12\sqrt{11}i}{-200}$$

Now solve the equation x=\frac{4±12\sqrt{11}i}{-200} when ± is plus. Add 4 to 12i\sqrt{11}.

$$x=\frac{4+12\sqrt{11}i}{-200}$$

Divide 4+12i\sqrt{11} by -200.

$$x=\frac{-3\sqrt{11}i-1}{50}$$

Now solve the equation x=\frac{4±12\sqrt{11}i}{-200} when ± is minus. Subtract 12i\sqrt{11} from 4.

$$x=\frac{-12\sqrt{11}i+4}{-200}$$

Divide 4-12i\sqrt{11} by -200.

$$x=\frac{-1+3\sqrt{11}i}{50}$$

The equation is now solved.

$$x=\frac{-3\sqrt{11}i-1}{50} x=\frac{-1+3\sqrt{11}i}{50}$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(14x+3\right)^{2}.

$$96x^{2}-\left(196x^{2}+84x+9\right)+80x=-5$$

To find the opposite of 196x^{2}+84x+9, find the opposite of each term.

$$96x^{2}-196x^{2}-84x-9+80x=-5$$

Combine 96x^{2} and -196x^{2} to get -100x^{2}.

$$-100x^{2}-84x-9+80x=-5$$

Combine -84x and 80x to get -4x.

$$-100x^{2}-4x-9=-5$$

Add 9 to both sides.

$$-100x^{2}-4x=-5+9$$

Add -5 and 9 to get 4.

$$-100x^{2}-4x=4$$

Divide both sides by -100.

$$\frac{-100x^{2}-4x}{-100}=\frac{4}{-100}$$

Dividing by -100 undoes the multiplication by -100.

$$x^{2}+\left(-\frac{4}{-100}\right)x=\frac{4}{-100}$$

Reduce the fraction \frac{-4}{-100} to lowest terms by extracting and canceling out 4.

$$x^{2}+\frac{1}{25}x=\frac{4}{-100}$$

Reduce the fraction \frac{4}{-100} to lowest terms by extracting and canceling out 4.

$$x^{2}+\frac{1}{25}x=-\frac{1}{25}$$

Divide \frac{1}{25}, the coefficient of the x term, by 2 to get \frac{1}{50}. Then add the square of \frac{1}{50} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{25}x+\left(\frac{1}{50}\right)^{2}=-\frac{1}{25}+\left(\frac{1}{50}\right)^{2}$$

Square \frac{1}{50} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{25}x+\frac{1}{2500}=-\frac{1}{25}+\frac{1}{2500}$$

Add -\frac{1}{25} to \frac{1}{2500} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{1}{25}x+\frac{1}{2500}=-\frac{99}{2500}$$

Factor x^{2}+\frac{1}{25}x+\frac{1}{2500}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{50}\right)^{2}=-\frac{99}{2500}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{50}\right)^{2}}=\sqrt{-\frac{99}{2500}}$$

Simplify.

$$x+\frac{1}{50}=\frac{3\sqrt{11}i}{50} x+\frac{1}{50}=-\frac{3\sqrt{11}i}{50}$$

Subtract \frac{1}{50} from both sides of the equation.

$$x=\frac{-1+3\sqrt{11}i}{50} x=\frac{-3\sqrt{11}i-1}{50}$$