Solve x^2/16+left(9xright)^/16+9x/4+frac{9/4}{12}=1

$$\frac{ { x }^{ 2 } }{ 16 } + \frac{ \frac{ { \left(9x \right) }^{ } }{ 16 } + \frac{ 9x }{ 4 } + \frac{ 9 }{ 4 } }{ 12 } = 1$$

Solve for x

$$x = \frac{\sqrt{1057} - 15}{8} \approx 2.188942052$$

$$x=\frac{-\sqrt{1057}-15}{8}\approx -5.938942052$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 48, the least common multiple of 16,12.

$$3x^{2}+4\left(\frac{\left(9x\right)^{1}}{16}+\frac{9x}{4}+\frac{9}{4}\right)=48$$

Calculate 9x to the power of 1 and get 9x.

$$3x^{2}+4\left(\frac{9x}{16}+\frac{9x}{4}+\frac{9}{4}\right)=48$$

Combine \frac{9x}{16} and \frac{9x}{4} to get \frac{5}{16}\times 9x.

$$3x^{2}+4\left(\frac{5}{16}\times 9x+\frac{9}{4}\right)=48$$

Use the distributive property to multiply 4 by \frac{5}{16}\times 9x+\frac{9}{4}.

$$3x^{2}+\frac{45}{4}x+9=48$$

Subtract 48 from both sides.

$$3x^{2}+\frac{45}{4}x+9-48=0$$

Subtract 48 from 9 to get -39.

$$3x^{2}+\frac{45}{4}x-39=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, \frac{45}{4} for b, and -39 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{45}{4}±\sqrt{\left(\frac{45}{4}\right)^{2}-4\times 3\left(-39\right)}}{2\times 3}$$

Square \frac{45}{4} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{45}{4}±\sqrt{\frac{2025}{16}-4\times 3\left(-39\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\frac{45}{4}±\sqrt{\frac{2025}{16}-12\left(-39\right)}}{2\times 3}$$

Multiply -12 times -39.

$$x=\frac{-\frac{45}{4}±\sqrt{\frac{2025}{16}+468}}{2\times 3}$$

Add \frac{2025}{16} to 468.

$$x=\frac{-\frac{45}{4}±\sqrt{\frac{9513}{16}}}{2\times 3}$$

Take the square root of \frac{9513}{16}.

$$x=\frac{-\frac{45}{4}±\frac{3\sqrt{1057}}{4}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-\frac{45}{4}±\frac{3\sqrt{1057}}{4}}{6}$$

Now solve the equation x=\frac{-\frac{45}{4}±\frac{3\sqrt{1057}}{4}}{6} when ± is plus. Add -\frac{45}{4} to \frac{3\sqrt{1057}}{4}.

$$x=\frac{3\sqrt{1057}-45}{4\times 6}$$

Divide \frac{-45+3\sqrt{1057}}{4} by 6.

$$x=\frac{\sqrt{1057}-15}{8}$$

Now solve the equation x=\frac{-\frac{45}{4}±\frac{3\sqrt{1057}}{4}}{6} when ± is minus. Subtract \frac{3\sqrt{1057}}{4} from -\frac{45}{4}.

$$x=\frac{-3\sqrt{1057}-45}{4\times 6}$$

Divide \frac{-45-3\sqrt{1057}}{4} by 6.

$$x=\frac{-\sqrt{1057}-15}{8}$$

The equation is now solved.

$$x=\frac{\sqrt{1057}-15}{8} x=\frac{-\sqrt{1057}-15}{8}$$

Steps for Completing the Square

Multiply both sides of the equation by 48, the least common multiple of 16,12.

$$3x^{2}+4\left(\frac{\left(9x\right)^{1}}{16}+\frac{9x}{4}+\frac{9}{4}\right)=48$$

Calculate 9x to the power of 1 and get 9x.

$$3x^{2}+4\left(\frac{9x}{16}+\frac{9x}{4}+\frac{9}{4}\right)=48$$

Combine \frac{9x}{16} and \frac{9x}{4} to get \frac{5}{16}\times 9x.

$$3x^{2}+4\left(\frac{5}{16}\times 9x+\frac{9}{4}\right)=48$$

Use the distributive property to multiply 4 by \frac{5}{16}\times 9x+\frac{9}{4}.

$$3x^{2}+\frac{45}{4}x+9=48$$

Subtract 9 from both sides.

$$3x^{2}+\frac{45}{4}x=48-9$$

Subtract 9 from 48 to get 39.

$$3x^{2}+\frac{45}{4}x=39$$

Divide both sides by 3.

$$\frac{3x^{2}+\frac{45}{4}x}{3}=\frac{39}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\frac{\frac{45}{4}}{3}x=\frac{39}{3}$$

Divide \frac{45}{4} by 3.

$$x^{2}+\frac{15}{4}x=\frac{39}{3}$$

Divide 39 by 3.

$$x^{2}+\frac{15}{4}x=13$$

Divide \frac{15}{4}, the coefficient of the x term, by 2 to get \frac{15}{8}. Then add the square of \frac{15}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{15}{4}x+\left(\frac{15}{8}\right)^{2}=13+\left(\frac{15}{8}\right)^{2}$$

Square \frac{15}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{15}{4}x+\frac{225}{64}=13+\frac{225}{64}$$

Add 13 to \frac{225}{64}.

$$x^{2}+\frac{15}{4}x+\frac{225}{64}=\frac{1057}{64}$$

Factor x^{2}+\frac{15}{4}x+\frac{225}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{15}{8}\right)^{2}=\frac{1057}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{15}{8}\right)^{2}}=\sqrt{\frac{1057}{64}}$$

Simplify.

$$x+\frac{15}{8}=\frac{\sqrt{1057}}{8} x+\frac{15}{8}=-\frac{\sqrt{1057}}{8}$$

Subtract \frac{15}{8} from both sides of the equation.

$$x=\frac{\sqrt{1057}-15}{8} x=\frac{-\sqrt{1057}-15}{8}$$