Solve 9d^2=289/4-52

$$9 { d }^{ 2 } = \frac{ 289 }{ 4 } -52$$

Solve for d

$$d = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

$$d = -\frac{3}{2} = -1\frac{1}{2} = -1.5$$

Steps Using Difference of Squares

Subtract 52 from \frac{289}{4} to get \frac{81}{4}.

$$9d^{2}=\frac{81}{4}$$

Subtract \frac{81}{4} from both sides.

$$9d^{2}-\frac{81}{4}=0$$

Divide both sides by \frac{9}{4}.

$$4d^{2}-9=0$$

Consider 4d^{2}-9. Rewrite 4d^{2}-9 as \left(2d\right)^{2}-3^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(2d-3\right)\left(2d+3\right)=0$$

To find equation solutions, solve 2d-3=0 and 2d+3=0.

$$d=\frac{3}{2} d=-\frac{3}{2}$$

Steps by Finding Square Root

Subtract 52 from \frac{289}{4} to get \frac{81}{4}.

$$9d^{2}=\frac{81}{4}$$

Divide both sides by 9.

$$d^{2}=\frac{\frac{81}{4}}{9}$$

Express \frac{\frac{81}{4}}{9} as a single fraction.

$$d^{2}=\frac{81}{4\times 9}$$

Multiply 4 and 9 to get 36.

$$d^{2}=\frac{81}{36}$$

Reduce the fraction \frac{81}{36} to lowest terms by extracting and canceling out 9.

$$d^{2}=\frac{9}{4}$$

Take the square root of both sides of the equation.

$$d=\frac{3}{2} d=-\frac{3}{2}$$

Steps Using the Quadratic Formula

Subtract 52 from \frac{289}{4} to get \frac{81}{4}.

$$9d^{2}=\frac{81}{4}$$

Subtract \frac{81}{4} from both sides.

$$9d^{2}-\frac{81}{4}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, 0 for b, and -\frac{81}{4} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$d=\frac{0±\sqrt{0^{2}-4\times 9\left(-\frac{81}{4}\right)}}{2\times 9}$$

Square 0.

$$d=\frac{0±\sqrt{-4\times 9\left(-\frac{81}{4}\right)}}{2\times 9}$$

Multiply -4 times 9.

$$d=\frac{0±\sqrt{-36\left(-\frac{81}{4}\right)}}{2\times 9}$$

Multiply -36 times -\frac{81}{4}.

$$d=\frac{0±\sqrt{729}}{2\times 9}$$

Take the square root of 729.

$$d=\frac{0±27}{2\times 9}$$

Multiply 2 times 9.

$$d=\frac{0±27}{18}$$

Now solve the equation d=\frac{0±27}{18} when ± is plus. Reduce the fraction \frac{27}{18} to lowest terms by extracting and canceling out 9.

$$d=\frac{3}{2}$$

Now solve the equation d=\frac{0±27}{18} when ± is minus. Reduce the fraction \frac{-27}{18} to lowest terms by extracting and canceling out 9.

$$d=-\frac{3}{2}$$

The equation is now solved.

$$d=\frac{3}{2} d=-\frac{3}{2}$$