Solve 14-(5x-1)(2x+3)=17-(10x+19(x-6)

$$14 - ( 5 x - 1 ) ( 2 x + 3 ) = 17 - ( 10 x + 19 ( x - 6 )$$

Solve for x (complex solution)

$$x=\frac{4+\sqrt{269}i}{5}\approx 0.8+3.280243893i$$

$$x=\frac{-\sqrt{269}i+4}{5}\approx 0.8-3.280243893i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 5x-1 by 2x+3 and combine like terms.

$$14-\left(10x^{2}+13x-3\right)=17-\left(10x+19\left(x-6\right)\right)$$

To find the opposite of 10x^{2}+13x-3, find the opposite of each term.

$$14-10x^{2}-13x+3=17-\left(10x+19\left(x-6\right)\right)$$

Add 14 and 3 to get 17.

$$17-10x^{2}-13x=17-\left(10x+19\left(x-6\right)\right)$$

Use the distributive property to multiply 19 by x-6.

$$17-10x^{2}-13x=17-\left(10x+19x-114\right)$$

Combine 10x and 19x to get 29x.

$$17-10x^{2}-13x=17-\left(29x-114\right)$$

To find the opposite of 29x-114, find the opposite of each term.

$$17-10x^{2}-13x=17-29x+114$$

Add 17 and 114 to get 131.

$$17-10x^{2}-13x=131-29x$$

Subtract 131 from both sides.

$$17-10x^{2}-13x-131=-29x$$

Subtract 131 from 17 to get -114.

$$-114-10x^{2}-13x=-29x$$

Add 29x to both sides.

$$-114-10x^{2}-13x+29x=0$$

Combine -13x and 29x to get 16x.

$$-114-10x^{2}+16x=0$$

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.

$$-10x^{2}+16x-114=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -10 for a, 16 for b, and -114 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\left(-10\right)\left(-114\right)}}{2\left(-10\right)}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\left(-10\right)\left(-114\right)}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$x=\frac{-16±\sqrt{256+40\left(-114\right)}}{2\left(-10\right)}$$

Multiply 40 times -114.

$$x=\frac{-16±\sqrt{256-4560}}{2\left(-10\right)}$$

Add 256 to -4560.

$$x=\frac{-16±\sqrt{-4304}}{2\left(-10\right)}$$

Take the square root of -4304.

$$x=\frac{-16±4\sqrt{269}i}{2\left(-10\right)}$$

Multiply 2 times -10.

$$x=\frac{-16±4\sqrt{269}i}{-20}$$

Now solve the equation x=\frac{-16±4\sqrt{269}i}{-20} when ± is plus. Add -16 to 4i\sqrt{269}.

$$x=\frac{-16+4\sqrt{269}i}{-20}$$

Divide -16+4i\sqrt{269} by -20.

$$x=\frac{-\sqrt{269}i+4}{5}$$

Now solve the equation x=\frac{-16±4\sqrt{269}i}{-20} when ± is minus. Subtract 4i\sqrt{269} from -16.

$$x=\frac{-4\sqrt{269}i-16}{-20}$$

Divide -16-4i\sqrt{269} by -20.

$$x=\frac{4+\sqrt{269}i}{5}$$

The equation is now solved.

$$x=\frac{-\sqrt{269}i+4}{5} x=\frac{4+\sqrt{269}i}{5}$$

Steps for Completing the Square

Use the distributive property to multiply 5x-1 by 2x+3 and combine like terms.

$$14-\left(10x^{2}+13x-3\right)=17-\left(10x+19\left(x-6\right)\right)$$

To find the opposite of 10x^{2}+13x-3, find the opposite of each term.

$$14-10x^{2}-13x+3=17-\left(10x+19\left(x-6\right)\right)$$

Add 14 and 3 to get 17.

$$17-10x^{2}-13x=17-\left(10x+19\left(x-6\right)\right)$$

Use the distributive property to multiply 19 by x-6.

$$17-10x^{2}-13x=17-\left(10x+19x-114\right)$$

Combine 10x and 19x to get 29x.

$$17-10x^{2}-13x=17-\left(29x-114\right)$$

To find the opposite of 29x-114, find the opposite of each term.

$$17-10x^{2}-13x=17-29x+114$$

Add 17 and 114 to get 131.

$$17-10x^{2}-13x=131-29x$$

Add 29x to both sides.

$$17-10x^{2}-13x+29x=131$$

Combine -13x and 29x to get 16x.

$$17-10x^{2}+16x=131$$

Subtract 17 from both sides.

$$-10x^{2}+16x=131-17$$

Subtract 17 from 131 to get 114.

$$-10x^{2}+16x=114$$

Divide both sides by -10.

$$\frac{-10x^{2}+16x}{-10}=\frac{114}{-10}$$

Dividing by -10 undoes the multiplication by -10.

$$x^{2}+\frac{16}{-10}x=\frac{114}{-10}$$

Reduce the fraction \frac{16}{-10} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{8}{5}x=\frac{114}{-10}$$

Reduce the fraction \frac{114}{-10} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{8}{5}x=-\frac{57}{5}$$

Divide -\frac{8}{5}, the coefficient of the x term, by 2 to get -\frac{4}{5}. Then add the square of -\frac{4}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{8}{5}x+\left(-\frac{4}{5}\right)^{2}=-\frac{57}{5}+\left(-\frac{4}{5}\right)^{2}$$

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

$$x^{2}-\frac{8}{5}x+\frac{16}{25}=-\frac{57}{5}+\frac{16}{25}$$

Add -\frac{57}{5} to \frac{16}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{8}{5}x+\frac{16}{25}=-\frac{269}{25}$$

Factor x^{2}-\frac{8}{5}x+\frac{16}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{4}{5}\right)^{2}=-\frac{269}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{4}{5}\right)^{2}}=\sqrt{-\frac{269}{25}}$$

Simplify.

$$x-\frac{4}{5}=\frac{\sqrt{269}i}{5} x-\frac{4}{5}=-\frac{\sqrt{269}i}{5}$$

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

$$x=\frac{4+\sqrt{269}i}{5} x=\frac{-\sqrt{269}i+4}{5}$$