Solve 1/2*47*(-x^2+2.25^2)=8.4

$$\frac { 1 } { 2 } \times 47 \cdot ( - x ^ { 2 } + 2.25 ^ { 2 } ) = 8.4$$

Solve for x

$$x = \frac{\sqrt{4157385}}{940} \approx 2.169113457$$

$$x = -\frac{\sqrt{4157385}}{940} \approx -2.169113457$$

Steps by Finding Square Root

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

$$\frac{47}{2}\left(-x^{2}+2.25^{2}\right)=8.4$$

Calculate 2.25 to the power of 2 and get 5.0625.

$$\frac{47}{2}\left(-x^{2}+5.0625\right)=8.4$$

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

$$\frac{47}{2}\left(-x^{2}\right)+\frac{3807}{32}=8.4$$

Subtract \frac{3807}{32} from both sides.

$$\frac{47}{2}\left(-x^{2}\right)=8.4-\frac{3807}{32}$$

Subtract \frac{3807}{32} from 8.4 to get -\frac{17691}{160}.

$$\frac{47}{2}\left(-x^{2}\right)=-\frac{17691}{160}$$

Multiply both sides by \frac{2}{47}, the reciprocal of \frac{47}{2}.

$$-x^{2}=-\frac{17691}{160}\times \frac{2}{47}$$

Multiply -\frac{17691}{160} and \frac{2}{47} to get -\frac{17691}{3760}.

$$-x^{2}=-\frac{17691}{3760}$$

Divide both sides by -1.

$$x^{2}=\frac{-\frac{17691}{3760}}{-1}$$

Express \frac{-\frac{17691}{3760}}{-1} as a single fraction.

$$x^{2}=\frac{-17691}{3760\left(-1\right)}$$

Multiply 3760 and -1 to get -3760.

$$x^{2}=\frac{-17691}{-3760}$$

Fraction \frac{-17691}{-3760} can be simplified to \frac{17691}{3760} by removing the negative sign from both the numerator and the denominator.

$$x^{2}=\frac{17691}{3760}$$

Take the square root of both sides of the equation.

$$x=\frac{\sqrt{4157385}}{940} x=-\frac{\sqrt{4157385}}{940}$$

Steps Using the Quadratic Formula

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

$$\frac{47}{2}\left(-x^{2}+2.25^{2}\right)=8.4$$

Calculate 2.25 to the power of 2 and get 5.0625.

$$\frac{47}{2}\left(-x^{2}+5.0625\right)=8.4$$

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

$$\frac{47}{2}\left(-x^{2}\right)+\frac{3807}{32}=8.4$$

Subtract 8.4 from both sides.

$$\frac{47}{2}\left(-x^{2}\right)+\frac{3807}{32}-8.4=0$$

Subtract 8.4 from \frac{3807}{32} to get \frac{17691}{160}.

$$\frac{47}{2}\left(-x^{2}\right)+\frac{17691}{160}=0$$

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

$$-\frac{47}{2}x^{2}+\frac{17691}{160}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{47}{2} for a, 0 for b, and \frac{17691}{160} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-\frac{47}{2}\right)\times \frac{17691}{160}}}{2\left(-\frac{47}{2}\right)}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-\frac{47}{2}\right)\times \frac{17691}{160}}}{2\left(-\frac{47}{2}\right)}$$

Multiply -4 times -\frac{47}{2}.

$$x=\frac{0±\sqrt{94\times \frac{17691}{160}}}{2\left(-\frac{47}{2}\right)}$$

Multiply 94 times \frac{17691}{160}.

$$x=\frac{0±\sqrt{\frac{831477}{80}}}{2\left(-\frac{47}{2}\right)}$$

Take the square root of \frac{831477}{80}.

$$x=\frac{0±\frac{\sqrt{4157385}}{20}}{2\left(-\frac{47}{2}\right)}$$

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

$$x=\frac{0±\frac{\sqrt{4157385}}{20}}{-47}$$

Now solve the equation x=\frac{0±\frac{\sqrt{4157385}}{20}}{-47} when ± is plus.

$$x=-\frac{\sqrt{4157385}}{940}$$

Now solve the equation x=\frac{0±\frac{\sqrt{4157385}}{20}}{-47} when ± is minus.

$$x=\frac{\sqrt{4157385}}{940}$$

The equation is now solved.

$$x=-\frac{\sqrt{4157385}}{940} x=\frac{\sqrt{4157385}}{940}$$