Solve -4=(1.4(x^2)-18)(2)+(2x-1)(2(1.8)x)/(2x-1)^2

$$- 4 = \frac { ( 1.4 ( x ^ { 2 } ) - 18 ) ( 2 ) + ( 2 x - 1 ) ( 2 ( 1.8 ) x ) } { ( 2 x - 1 ) ^ { 2 } }$$

Solve for x

$$x = \frac{\sqrt{23201} + 49}{130} \approx 1.548605729$$

$$x=\frac{49-\sqrt{23201}}{130}\approx -0.794759575$$

Steps Using the Quadratic Formula

Variable x cannot be equal to \frac{1}{2} since division by zero is not defined. Multiply both sides of the equation by \left(2x-1\right)^{2}.

$$-4\left(2x-1\right)^{2}=\left(1.4x^{2}-18\right)\times 2+\left(2x-1\right)\times 2\times 1.8x$$

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

$$-4\left(4x^{2}-4x+1\right)=\left(1.4x^{2}-18\right)\times 2+\left(2x-1\right)\times 2\times 1.8x$$

Use the distributive property to multiply -4 by 4x^{2}-4x+1.

$$-16x^{2}+16x-4=\left(1.4x^{2}-18\right)\times 2+\left(2x-1\right)\times 2\times 1.8x$$

Use the distributive property to multiply 1.4x^{2}-18 by 2.

$$-16x^{2}+16x-4=2.8x^{2}-36+\left(2x-1\right)\times 2\times 1.8x$$

Multiply 2 and 1.8 to get 3.6.

$$-16x^{2}+16x-4=2.8x^{2}-36+\left(2x-1\right)\times 3.6x$$

Use the distributive property to multiply 2x-1 by 3.6.

$$-16x^{2}+16x-4=2.8x^{2}-36+\left(7.2x-3.6\right)x$$

Use the distributive property to multiply 7.2x-3.6 by x.

$$-16x^{2}+16x-4=2.8x^{2}-36+7.2x^{2}-3.6x$$

Combine 2.8x^{2} and 7.2x^{2} to get 10x^{2}.

$$-16x^{2}+16x-4=10x^{2}-36-3.6x$$

Subtract 10x^{2} from both sides.

$$-16x^{2}+16x-4-10x^{2}=-36-3.6x$$

Combine -16x^{2} and -10x^{2} to get -26x^{2}.

$$-26x^{2}+16x-4=-36-3.6x$$

Subtract -36 from both sides.

$$-26x^{2}+16x-4-\left(-36\right)=-3.6x$$

The opposite of -36 is 36.

$$-26x^{2}+16x-4+36=-3.6x$$

Add 3.6x to both sides.

$$-26x^{2}+16x-4+36+3.6x=0$$

Add -4 and 36 to get 32.

$$-26x^{2}+16x+32+3.6x=0$$

Combine 16x and 3.6x to get 19.6x.

$$-26x^{2}+19.6x+32=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -26 for a, 19.6 for b, and 32 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-19.6±\sqrt{19.6^{2}-4\left(-26\right)\times 32}}{2\left(-26\right)}$$

Square 19.6 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-19.6±\sqrt{384.16-4\left(-26\right)\times 32}}{2\left(-26\right)}$$

Multiply -4 times -26.

$$x=\frac{-19.6±\sqrt{384.16+104\times 32}}{2\left(-26\right)}$$

Multiply 104 times 32.

$$x=\frac{-19.6±\sqrt{384.16+3328}}{2\left(-26\right)}$$

Add 384.16 to 3328.

$$x=\frac{-19.6±\sqrt{3712.16}}{2\left(-26\right)}$$

Take the square root of 3712.16.

$$x=\frac{-19.6±\frac{2\sqrt{23201}}{5}}{2\left(-26\right)}$$

Multiply 2 times -26.

$$x=\frac{-19.6±\frac{2\sqrt{23201}}{5}}{-52}$$

Now solve the equation x=\frac{-19.6±\frac{2\sqrt{23201}}{5}}{-52} when ± is plus. Add -19.6 to \frac{2\sqrt{23201}}{5}.

$$x=\frac{2\sqrt{23201}-98}{-52\times 5}$$

Divide \frac{-98+2\sqrt{23201}}{5} by -52.

$$x=\frac{49-\sqrt{23201}}{130}$$

Now solve the equation x=\frac{-19.6±\frac{2\sqrt{23201}}{5}}{-52} when ± is minus. Subtract \frac{2\sqrt{23201}}{5} from -19.6.

$$x=\frac{-2\sqrt{23201}-98}{-52\times 5}$$

Divide \frac{-98-2\sqrt{23201}}{5} by -52.

$$x=\frac{\sqrt{23201}+49}{130}$$

The equation is now solved.

$$x=\frac{49-\sqrt{23201}}{130} x=\frac{\sqrt{23201}+49}{130}$$

Steps for Completing the Square

Variable x cannot be equal to \frac{1}{2} since division by zero is not defined. Multiply both sides of the equation by \left(2x-1\right)^{2}.

$$-4\left(2x-1\right)^{2}=\left(1.4x^{2}-18\right)\times 2+\left(2x-1\right)\times 2\times 1.8x$$

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

$$-4\left(4x^{2}-4x+1\right)=\left(1.4x^{2}-18\right)\times 2+\left(2x-1\right)\times 2\times 1.8x$$

Use the distributive property to multiply -4 by 4x^{2}-4x+1.

$$-16x^{2}+16x-4=\left(1.4x^{2}-18\right)\times 2+\left(2x-1\right)\times 2\times 1.8x$$

Use the distributive property to multiply 1.4x^{2}-18 by 2.

$$-16x^{2}+16x-4=2.8x^{2}-36+\left(2x-1\right)\times 2\times 1.8x$$

Multiply 2 and 1.8 to get 3.6.

$$-16x^{2}+16x-4=2.8x^{2}-36+\left(2x-1\right)\times 3.6x$$

Use the distributive property to multiply 2x-1 by 3.6.

$$-16x^{2}+16x-4=2.8x^{2}-36+\left(7.2x-3.6\right)x$$

Use the distributive property to multiply 7.2x-3.6 by x.

$$-16x^{2}+16x-4=2.8x^{2}-36+7.2x^{2}-3.6x$$

Combine 2.8x^{2} and 7.2x^{2} to get 10x^{2}.

$$-16x^{2}+16x-4=10x^{2}-36-3.6x$$

Subtract 10x^{2} from both sides.

$$-16x^{2}+16x-4-10x^{2}=-36-3.6x$$

Combine -16x^{2} and -10x^{2} to get -26x^{2}.

$$-26x^{2}+16x-4=-36-3.6x$$

Add 3.6x to both sides.

$$-26x^{2}+16x-4+3.6x=-36$$

Combine 16x and 3.6x to get 19.6x.

$$-26x^{2}+19.6x-4=-36$$

Add 4 to both sides.

$$-26x^{2}+19.6x=-36+4$$

Add -36 and 4 to get -32.

$$-26x^{2}+19.6x=-32$$

Divide both sides by -26.

$$\frac{-26x^{2}+19.6x}{-26}=-\frac{32}{-26}$$

Dividing by -26 undoes the multiplication by -26.

$$x^{2}+\frac{19.6}{-26}x=-\frac{32}{-26}$$

Divide 19.6 by -26.

$$x^{2}-\frac{49}{65}x=-\frac{32}{-26}$$

Reduce the fraction \frac{-32}{-26} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{49}{65}x=\frac{16}{13}$$

Divide -\frac{49}{65}, the coefficient of the x term, by 2 to get -\frac{49}{130}. Then add the square of -\frac{49}{130} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{49}{65}x+\left(-\frac{49}{130}\right)^{2}=\frac{16}{13}+\left(-\frac{49}{130}\right)^{2}$$

Square -\frac{49}{130} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{49}{65}x+\frac{2401}{16900}=\frac{16}{13}+\frac{2401}{16900}$$

Add \frac{16}{13} to \frac{2401}{16900} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{49}{65}x+\frac{2401}{16900}=\frac{23201}{16900}$$

Factor x^{2}-\frac{49}{65}x+\frac{2401}{16900}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{49}{130}\right)^{2}=\frac{23201}{16900}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{49}{130}\right)^{2}}=\sqrt{\frac{23201}{16900}}$$

Simplify.

$$x-\frac{49}{130}=\frac{\sqrt{23201}}{130} x-\frac{49}{130}=-\frac{\sqrt{23201}}{130}$$

Add \frac{49}{130} to both sides of the equation.

$$x=\frac{\sqrt{23201}+49}{130} x=\frac{49-\sqrt{23201}}{130}$$