Solve 18x^2+66x-126/17-3x-6x-18=0

$$\frac { 18 x ^ { 2 } + 66 x - 126 } { 17 - 3 x } - 6 x - 18 = 0$$

Solve for x

$$x = \frac{\sqrt{193} - 1}{4} \approx 3.223110997$$

$$x=\frac{-\sqrt{193}-1}{4}\approx -3.723110997$$

Steps Using the Quadratic Formula

Variable x cannot be equal to \frac{17}{3} since division by zero is not defined. Multiply both sides of the equation by -3x+17.

$$18x^{2}+66x-126-6x\left(-3x+17\right)+\left(-3x+17\right)\left(-18\right)=0$$

Use the distributive property to multiply -6x by -3x+17.

$$18x^{2}+66x-126+18x^{2}-102x+\left(-3x+17\right)\left(-18\right)=0$$

Combine 18x^{2} and 18x^{2} to get 36x^{2}.

$$36x^{2}+66x-126-102x+\left(-3x+17\right)\left(-18\right)=0$$

Combine 66x and -102x to get -36x.

$$36x^{2}-36x-126+\left(-3x+17\right)\left(-18\right)=0$$

Use the distributive property to multiply -3x+17 by -18.

$$36x^{2}-36x-126+54x-306=0$$

Combine -36x and 54x to get 18x.

$$36x^{2}+18x-126-306=0$$

Subtract 306 from -126 to get -432.

$$36x^{2}+18x-432=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 36 for a, 18 for b, and -432 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-18±\sqrt{18^{2}-4\times 36\left(-432\right)}}{2\times 36}$$

Square 18.

$$x=\frac{-18±\sqrt{324-4\times 36\left(-432\right)}}{2\times 36}$$

Multiply -4 times 36.

$$x=\frac{-18±\sqrt{324-144\left(-432\right)}}{2\times 36}$$

Multiply -144 times -432.

$$x=\frac{-18±\sqrt{324+62208}}{2\times 36}$$

Add 324 to 62208.

$$x=\frac{-18±\sqrt{62532}}{2\times 36}$$

Take the square root of 62532.

$$x=\frac{-18±18\sqrt{193}}{2\times 36}$$

Multiply 2 times 36.

$$x=\frac{-18±18\sqrt{193}}{72}$$

Now solve the equation x=\frac{-18±18\sqrt{193}}{72} when ± is plus. Add -18 to 18\sqrt{193}.

$$x=\frac{18\sqrt{193}-18}{72}$$

Divide -18+18\sqrt{193} by 72.

$$x=\frac{\sqrt{193}-1}{4}$$

Now solve the equation x=\frac{-18±18\sqrt{193}}{72} when ± is minus. Subtract 18\sqrt{193} from -18.

$$x=\frac{-18\sqrt{193}-18}{72}$$

Divide -18-18\sqrt{193} by 72.

$$x=\frac{-\sqrt{193}-1}{4}$$

The equation is now solved.

$$x=\frac{\sqrt{193}-1}{4} x=\frac{-\sqrt{193}-1}{4}$$

Steps for Completing the Square

Variable x cannot be equal to \frac{17}{3} since division by zero is not defined. Multiply both sides of the equation by -3x+17.

$$18x^{2}+66x-126-6x\left(-3x+17\right)+\left(-3x+17\right)\left(-18\right)=0$$

Use the distributive property to multiply -6x by -3x+17.

$$18x^{2}+66x-126+18x^{2}-102x+\left(-3x+17\right)\left(-18\right)=0$$

Combine 18x^{2} and 18x^{2} to get 36x^{2}.

$$36x^{2}+66x-126-102x+\left(-3x+17\right)\left(-18\right)=0$$

Combine 66x and -102x to get -36x.

$$36x^{2}-36x-126+\left(-3x+17\right)\left(-18\right)=0$$

Use the distributive property to multiply -3x+17 by -18.

$$36x^{2}-36x-126+54x-306=0$$

Combine -36x and 54x to get 18x.

$$36x^{2}+18x-126-306=0$$

Subtract 306 from -126 to get -432.

$$36x^{2}+18x-432=0$$

Add 432 to both sides. Anything plus zero gives itself.

$$36x^{2}+18x=432$$

Divide both sides by 36.

$$\frac{36x^{2}+18x}{36}=\frac{432}{36}$$

Dividing by 36 undoes the multiplication by 36.

$$x^{2}+\frac{18}{36}x=\frac{432}{36}$$

Reduce the fraction \frac{18}{36} to lowest terms by extracting and canceling out 18.

$$x^{2}+\frac{1}{2}x=\frac{432}{36}$$

Divide 432 by 36.

$$x^{2}+\frac{1}{2}x=12$$

Divide \frac{1}{2}, the coefficient of the x term, by 2 to get \frac{1}{4}. Then add the square of \frac{1}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{2}x+\left(\frac{1}{4}\right)^{2}=12+\left(\frac{1}{4}\right)^{2}$$

Square \frac{1}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{2}x+\frac{1}{16}=12+\frac{1}{16}$$

Add 12 to \frac{1}{16}.

$$x^{2}+\frac{1}{2}x+\frac{1}{16}=\frac{193}{16}$$

Factor x^{2}+\frac{1}{2}x+\frac{1}{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}.

$$\left(x+\frac{1}{4}\right)^{2}=\frac{193}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{4}\right)^{2}}=\sqrt{\frac{193}{16}}$$

Simplify.

$$x+\frac{1}{4}=\frac{\sqrt{193}}{4} x+\frac{1}{4}=-\frac{\sqrt{193}}{4}$$

Subtract \frac{1}{4} from both sides of the equation.

$$x=\frac{\sqrt{193}-1}{4} x=\frac{-\sqrt{193}-1}{4}$$