Solve 1/24.7left(x^2-2.25^2right)=8.42

$$\frac{ 1 }{ 2 } 4.7 \left( { x }^{ 2 } - { 2.25 }^{ 2 } \right) = 8.42$$

Solve for x

$$x = \frac{\sqrt{7639145}}{940} \approx 2.940319493$$

$$x = -\frac{\sqrt{7639145}}{940} \approx -2.940319493$$

Steps by Finding Square Root

Multiply \frac{1}{2} and 4.7 to get \frac{47}{20}.

$$\frac{47}{20}\left(x^{2}-2.25^{2}\right)=8.42$$

Calculate 2.25 to the power of 2 and get 5.0625.

$$\frac{47}{20}\left(x^{2}-5.0625\right)=8.42$$

Use the distributive property to multiply \frac{47}{20} by x^{2}-5.0625.

$$\frac{47}{20}x^{2}-\frac{3807}{320}=8.42$$

Add \frac{3807}{320} to both sides.

$$\frac{47}{20}x^{2}=8.42+\frac{3807}{320}$$

Add 8.42 and \frac{3807}{320} to get \frac{32507}{1600}.

$$\frac{47}{20}x^{2}=\frac{32507}{1600}$$

Multiply both sides by \frac{20}{47}, the reciprocal of \frac{47}{20}.

$$x^{2}=\frac{32507}{1600}\times \frac{20}{47}$$

Multiply \frac{32507}{1600} and \frac{20}{47} to get \frac{32507}{3760}.

$$x^{2}=\frac{32507}{3760}$$

Take the square root of both sides of the equation.

$$x=\frac{\sqrt{7639145}}{940} x=-\frac{\sqrt{7639145}}{940}$$

Steps Using the Quadratic Formula

Multiply \frac{1}{2} and 4.7 to get \frac{47}{20}.

$$\frac{47}{20}\left(x^{2}-2.25^{2}\right)=8.42$$

Calculate 2.25 to the power of 2 and get 5.0625.

$$\frac{47}{20}\left(x^{2}-5.0625\right)=8.42$$

Use the distributive property to multiply \frac{47}{20} by x^{2}-5.0625.

$$\frac{47}{20}x^{2}-\frac{3807}{320}=8.42$$

Subtract 8.42 from both sides.

$$\frac{47}{20}x^{2}-\frac{3807}{320}-8.42=0$$

Subtract 8.42 from -\frac{3807}{320} to get -\frac{32507}{1600}.

$$\frac{47}{20}x^{2}-\frac{32507}{1600}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{47}{20} for a, 0 for b, and -\frac{32507}{1600} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times \frac{47}{20}\left(-\frac{32507}{1600}\right)}}{2\times \frac{47}{20}}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times \frac{47}{20}\left(-\frac{32507}{1600}\right)}}{2\times \frac{47}{20}}$$

Multiply -4 times \frac{47}{20}.

$$x=\frac{0±\sqrt{-\frac{47}{5}\left(-\frac{32507}{1600}\right)}}{2\times \frac{47}{20}}$$

Multiply -\frac{47}{5} times -\frac{32507}{1600} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{0±\sqrt{\frac{1527829}{8000}}}{2\times \frac{47}{20}}$$

Take the square root of \frac{1527829}{8000}.

$$x=\frac{0±\frac{\sqrt{7639145}}{200}}{2\times \frac{47}{20}}$$

Multiply 2 times \frac{47}{20}.

$$x=\frac{0±\frac{\sqrt{7639145}}{200}}{\frac{47}{10}}$$

Now solve the equation x=\frac{0±\frac{\sqrt{7639145}}{200}}{\frac{47}{10}} when ± is plus.

$$x=\frac{\sqrt{7639145}}{940}$$

Now solve the equation x=\frac{0±\frac{\sqrt{7639145}}{200}}{\frac{47}{10}} when ± is minus.

$$x=-\frac{\sqrt{7639145}}{940}$$

The equation is now solved.

$$x=\frac{\sqrt{7639145}}{940} x=-\frac{\sqrt{7639145}}{940}$$