Solve x^2-x^2/4-x/4*3x/4-x^2/8=frac{21}{4}

$${ x }^{ 2 } - \frac{ { x }^{ 2 } }{ 4 } - \frac{ x }{ 4 } \times \frac{ 3x }{ 4 } - \frac{ { x }^{ 2 } }{ 8 } = \frac{ 21 }{ 4 }$$

Solve for x

$$x=2\sqrt{3}\approx 3.464101615$$

$$x=-2\sqrt{3}\approx -3.464101615$$

Steps by Finding Square Root

Multiply both sides of the equation by 8, the least common multiple of 4,8.

$$8\left(x^{2}-\frac{x^{2}}{4}-\frac{x}{4}\times \frac{3x}{4}\right)-x^{2}=42$$

Combine x^{2} and -\frac{x^{2}}{4} to get \frac{3}{4}x^{2}.

$$8\left(\frac{3}{4}x^{2}-\frac{x}{4}\times \frac{3x}{4}\right)-x^{2}=42$$

Multiply \frac{x}{4} times \frac{3x}{4} by multiplying numerator times numerator and denominator times denominator.

$$8\left(\frac{3}{4}x^{2}-\frac{x\times 3x}{4\times 4}\right)-x^{2}=42$$

Multiply x and x to get x^{2}.

$$8\left(\frac{3}{4}x^{2}-\frac{x^{2}\times 3}{4\times 4}\right)-x^{2}=42$$

Multiply 4 and 4 to get 16.

$$8\left(\frac{3}{4}x^{2}-\frac{x^{2}\times 3}{16}\right)-x^{2}=42$$

Use the distributive property to multiply 8 by \frac{3}{4}x^{2}-\frac{x^{2}\times 3}{16}.

$$6x^{2}+8\left(-\frac{x^{2}\times 3}{16}\right)-x^{2}=42$$

Cancel out 16, the greatest common factor in 8 and 16.

$$6x^{2}+\frac{x^{2}\times 3}{-2}-x^{2}=42$$

Multiply both sides of the equation by -2.

$$-12x^{2}+x^{2}\times 3+2x^{2}=-84$$

Combine -12x^{2} and x^{2}\times 3 to get -9x^{2}.

$$-9x^{2}+2x^{2}=-84$$

Combine -9x^{2} and 2x^{2} to get -7x^{2}.

$$-7x^{2}=-84$$

Divide both sides by -7.

$$x^{2}=\frac{-84}{-7}$$

Divide -84 by -7 to get 12.

$$x^{2}=12$$

Take the square root of both sides of the equation.

$$x=2\sqrt{3} x=-2\sqrt{3}$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 8, the least common multiple of 4,8.

$$8\left(x^{2}-\frac{x^{2}}{4}-\frac{x}{4}\times \frac{3x}{4}\right)-x^{2}=42$$

Combine x^{2} and -\frac{x^{2}}{4} to get \frac{3}{4}x^{2}.

$$8\left(\frac{3}{4}x^{2}-\frac{x}{4}\times \frac{3x}{4}\right)-x^{2}=42$$

Multiply \frac{x}{4} times \frac{3x}{4} by multiplying numerator times numerator and denominator times denominator.

$$8\left(\frac{3}{4}x^{2}-\frac{x\times 3x}{4\times 4}\right)-x^{2}=42$$

Multiply x and x to get x^{2}.

$$8\left(\frac{3}{4}x^{2}-\frac{x^{2}\times 3}{4\times 4}\right)-x^{2}=42$$

Multiply 4 and 4 to get 16.

$$8\left(\frac{3}{4}x^{2}-\frac{x^{2}\times 3}{16}\right)-x^{2}=42$$

Use the distributive property to multiply 8 by \frac{3}{4}x^{2}-\frac{x^{2}\times 3}{16}.

$$6x^{2}+8\left(-\frac{x^{2}\times 3}{16}\right)-x^{2}=42$$

Cancel out 16, the greatest common factor in 8 and 16.

$$6x^{2}+\frac{x^{2}\times 3}{-2}-x^{2}=42$$

Subtract 42 from both sides.

$$6x^{2}+\frac{x^{2}\times 3}{-2}-x^{2}-42=0$$

Multiply both sides of the equation by -2.

$$-12x^{2}+x^{2}\times 3+2x^{2}+84=0$$

Combine -12x^{2} and x^{2}\times 3 to get -9x^{2}.

$$-9x^{2}+2x^{2}+84=0$$

Combine -9x^{2} and 2x^{2} to get -7x^{2}.

$$-7x^{2}+84=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -7 for a, 0 for b, and 84 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-7\right)\times 84}}{2\left(-7\right)}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-7\right)\times 84}}{2\left(-7\right)}$$

Multiply -4 times -7.

$$x=\frac{0±\sqrt{28\times 84}}{2\left(-7\right)}$$

Multiply 28 times 84.

$$x=\frac{0±\sqrt{2352}}{2\left(-7\right)}$$

Take the square root of 2352.

$$x=\frac{0±28\sqrt{3}}{2\left(-7\right)}$$

Multiply 2 times -7.

$$x=\frac{0±28\sqrt{3}}{-14}$$

Now solve the equation x=\frac{0±28\sqrt{3}}{-14} when ± is plus.

$$x=-2\sqrt{3}$$

Now solve the equation x=\frac{0±28\sqrt{3}}{-14} when ± is minus.

$$x=2\sqrt{3}$$

The equation is now solved.

$$x=-2\sqrt{3} x=2\sqrt{3}$$