Solve 8(12x-10)-5(19x^2-1)/99x=1

$$8 ( 12 x - 10 ) - \frac { 5 ( 19 x ^ { 2 } - 1 ) } { 99 x } = 1$$

Solve for x

$$x=\frac{\sqrt{64116181}+8019}{18818}\approx 0.851645128$$

$$x=\frac{8019-\sqrt{64116181}}{18818}\approx 0.000623976$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 99x.

$$8\left(12x-10\right)\times 99x-5\left(19x^{2}-1\right)=99x$$

Multiply 8 and 99 to get 792.

$$792\left(12x-10\right)x-5\left(19x^{2}-1\right)=99x$$

Use the distributive property to multiply 792 by 12x-10.

$$\left(9504x-7920\right)x-5\left(19x^{2}-1\right)=99x$$

Use the distributive property to multiply 9504x-7920 by x.

$$9504x^{2}-7920x-5\left(19x^{2}-1\right)=99x$$

Use the distributive property to multiply -5 by 19x^{2}-1.

$$9504x^{2}-7920x-95x^{2}+5=99x$$

Combine 9504x^{2} and -95x^{2} to get 9409x^{2}.

$$9409x^{2}-7920x+5=99x$$

Subtract 99x from both sides.

$$9409x^{2}-7920x+5-99x=0$$

Combine -7920x and -99x to get -8019x.

$$9409x^{2}-8019x+5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9409 for a, -8019 for b, and 5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8019\right)±\sqrt{\left(-8019\right)^{2}-4\times 9409\times 5}}{2\times 9409}$$

Square -8019.

$$x=\frac{-\left(-8019\right)±\sqrt{64304361-4\times 9409\times 5}}{2\times 9409}$$

Multiply -4 times 9409.

$$x=\frac{-\left(-8019\right)±\sqrt{64304361-37636\times 5}}{2\times 9409}$$

Multiply -37636 times 5.

$$x=\frac{-\left(-8019\right)±\sqrt{64304361-188180}}{2\times 9409}$$

Add 64304361 to -188180.

$$x=\frac{-\left(-8019\right)±\sqrt{64116181}}{2\times 9409}$$

The opposite of -8019 is 8019.

$$x=\frac{8019±\sqrt{64116181}}{2\times 9409}$$

Multiply 2 times 9409.

$$x=\frac{8019±\sqrt{64116181}}{18818}$$

Now solve the equation x=\frac{8019±\sqrt{64116181}}{18818} when ± is plus. Add 8019 to \sqrt{64116181}.

$$x=\frac{\sqrt{64116181}+8019}{18818}$$

Now solve the equation x=\frac{8019±\sqrt{64116181}}{18818} when ± is minus. Subtract \sqrt{64116181} from 8019.

$$x=\frac{8019-\sqrt{64116181}}{18818}$$

The equation is now solved.

$$x=\frac{\sqrt{64116181}+8019}{18818} x=\frac{8019-\sqrt{64116181}}{18818}$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 99x.

$$8\left(12x-10\right)\times 99x-5\left(19x^{2}-1\right)=99x$$

Multiply 8 and 99 to get 792.

$$792\left(12x-10\right)x-5\left(19x^{2}-1\right)=99x$$

Use the distributive property to multiply 792 by 12x-10.

$$\left(9504x-7920\right)x-5\left(19x^{2}-1\right)=99x$$

Use the distributive property to multiply 9504x-7920 by x.

$$9504x^{2}-7920x-5\left(19x^{2}-1\right)=99x$$

Use the distributive property to multiply -5 by 19x^{2}-1.

$$9504x^{2}-7920x-95x^{2}+5=99x$$

Combine 9504x^{2} and -95x^{2} to get 9409x^{2}.

$$9409x^{2}-7920x+5=99x$$

Subtract 99x from both sides.

$$9409x^{2}-7920x+5-99x=0$$

Combine -7920x and -99x to get -8019x.

$$9409x^{2}-8019x+5=0$$

Subtract 5 from both sides. Anything subtracted from zero gives its negation.

$$9409x^{2}-8019x=-5$$

Divide both sides by 9409.

$$\frac{9409x^{2}-8019x}{9409}=-\frac{5}{9409}$$

Dividing by 9409 undoes the multiplication by 9409.

$$x^{2}-\frac{8019}{9409}x=-\frac{5}{9409}$$

Divide -\frac{8019}{9409}, the coefficient of the x term, by 2 to get -\frac{8019}{18818}. Then add the square of -\frac{8019}{18818} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{8019}{9409}x+\left(-\frac{8019}{18818}\right)^{2}=-\frac{5}{9409}+\left(-\frac{8019}{18818}\right)^{2}$$

Square -\frac{8019}{18818} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{8019}{9409}x+\frac{64304361}{354117124}=-\frac{5}{9409}+\frac{64304361}{354117124}$$

Add -\frac{5}{9409} to \frac{64304361}{354117124} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{8019}{9409}x+\frac{64304361}{354117124}=\frac{64116181}{354117124}$$

Factor x^{2}-\frac{8019}{9409}x+\frac{64304361}{354117124}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{8019}{18818}\right)^{2}=\frac{64116181}{354117124}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{8019}{18818}\right)^{2}}=\sqrt{\frac{64116181}{354117124}}$$

Simplify.

$$x-\frac{8019}{18818}=\frac{\sqrt{64116181}}{18818} x-\frac{8019}{18818}=-\frac{\sqrt{64116181}}{18818}$$

Add \frac{8019}{18818} to both sides of the equation.

$$x=\frac{\sqrt{64116181}+8019}{18818} x=\frac{8019-\sqrt{64116181}}{18818}$$