Pre-Algebra Examples

$d^{2} + {(d - 15)}^{2} = 900$

Step 1

Move all terms to the left side of the equation and simplify.

Tap for more steps...

Step 1.1

Subtract $900$ from both sides of the equation.

$d^{2} + {(d - 15)}^{2} - 900 = 0$

Step 1.2

Simplify $d^{2} + {(d - 15)}^{2} - 900$ .

Tap for more steps...

Step 1.2.1

Simplify each term.

Tap for more steps...

Step 1.2.1.1

Rewrite ${(d - 15)}^{2}$ as $(d - 15) (d - 15)$ .

$d^{2} + (d - 15) (d - 15) - 900 = 0$

Step 1.2.1.2

Expand $(d - 15) (d - 15)$ using the FOIL Method.

Tap for more steps...

Step 1.2.1.2.1

Apply the distributive property.

$d^{2} + d (d - 15) - 15 (d - 15) - 900 = 0$

Step 1.2.1.2.2

Apply the distributive property.

$d^{2} + d \cdot d + d \cdot - 15 - 15 (d - 15) - 900 = 0$

Step 1.2.1.2.3

Apply the distributive property.

$d^{2} + d \cdot d + d \cdot - 15 - 15 d - 15 \cdot - 15 - 900 = 0$

Step 1.2.1.3

Simplify and combine like terms.

Tap for more steps...

Step 1.2.1.3.1

Simplify each term.

Tap for more steps...

Step 1.2.1.3.1.1

Multiply $d$ by $d$ .

$d^{2} + d^{2} + d \cdot - 15 - 15 d - 15 \cdot - 15 - 900 = 0$

Step 1.2.1.3.1.2

Move $- 15$ to the left of $d$ .

$d^{2} + d^{2} - 15 \cdot d - 15 d - 15 \cdot - 15 - 900 = 0$

Step 1.2.1.3.1.3

Multiply $- 15$ by $- 15$ .

$d^{2} + d^{2} - 15 d - 15 d + 225 - 900 = 0$

Step 1.2.1.3.2

Subtract $15 d$ from $- 15 d$ .

$d^{2} + d^{2} - 30 d + 225 - 900 = 0$

Step 1.2.2

Add $d^{2}$ and $d^{2}$ .

$2 d^{2} - 30 d + 225 - 900 = 0$

Step 1.2.3

Subtract $900$ from $225$ .

$2 d^{2} - 30 d - 675 = 0$

Step 2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3

Substitute the values $a = 2$ , $b = - 30$ , and $c = - 675$ into the quadratic formula and solve for $d$ .

$\frac{30 \pm \sqrt{{(- 30)}^{2} - 4 \cdot (2 \cdot - 675)}}{2 \cdot 2}$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Simplify the numerator.

Tap for more steps...

Step 4.1.1

Raise $- 30$ to the power of $2$ .

$d = \frac{30 \pm \sqrt{900 - 4 \cdot 2 \cdot - 675}}{2 \cdot 2}$

Step 4.1.2

Multiply $- 4 \cdot 2 \cdot - 675$ .

Tap for more steps...

Step 4.1.2.1

Multiply $- 4$ by $2$ .

$d = \frac{30 \pm \sqrt{900 - 8 \cdot - 675}}{2 \cdot 2}$

Step 4.1.2.2

Multiply $- 8$ by $- 675$ .

$d = \frac{30 \pm \sqrt{900 + 5400}}{2 \cdot 2}$

Step 4.1.3

Add $900$ and $5400$ .

$d = \frac{30 \pm \sqrt{6300}}{2 \cdot 2}$

Step 4.1.4

Rewrite $6300$ as $30^{2} \cdot 7$ .

Tap for more steps...

Step 4.1.4.1

Factor $900$ out of $6300$ .

$d = \frac{30 \pm \sqrt{900 (7)}}{2 \cdot 2}$

Step 4.1.4.2

Rewrite $900$ as $30^{2}$ .

$d = \frac{30 \pm \sqrt{30^{2} \cdot 7}}{2 \cdot 2}$

Step 4.1.5

Pull terms out from under the radical.

$d = \frac{30 \pm 30 \sqrt{7}}{2 \cdot 2}$

Step 4.2

Multiply $2$ by $2$ .

$d = \frac{30 \pm 30 \sqrt{7}}{4}$

Step 4.3

Simplify $\frac{30 \pm 30 \sqrt{7}}{4}$ .

$d = \frac{15 \pm 15 \sqrt{7}}{2}$

Step 5

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Tap for more steps...

Step 5.1

Simplify the numerator.

Tap for more steps...

Step 5.1.1

Raise $- 30$ to the power of $2$ .

$d = \frac{30 \pm \sqrt{900 - 4 \cdot 2 \cdot - 675}}{2 \cdot 2}$

Step 5.1.2

Multiply $- 4 \cdot 2 \cdot - 675$ .

Tap for more steps...

Step 5.1.2.1

Multiply $- 4$ by $2$ .

$d = \frac{30 \pm \sqrt{900 - 8 \cdot - 675}}{2 \cdot 2}$

Step 5.1.2.2

Multiply $- 8$ by $- 675$ .

$d = \frac{30 \pm \sqrt{900 + 5400}}{2 \cdot 2}$

Step 5.1.3

Add $900$ and $5400$ .

$d = \frac{30 \pm \sqrt{6300}}{2 \cdot 2}$

Step 5.1.4

Rewrite $6300$ as $30^{2} \cdot 7$ .

Tap for more steps...

Step 5.1.4.1

Factor $900$ out of $6300$ .

$d = \frac{30 \pm \sqrt{900 (7)}}{2 \cdot 2}$

Step 5.1.4.2

Rewrite $900$ as $30^{2}$ .

$d = \frac{30 \pm \sqrt{30^{2} \cdot 7}}{2 \cdot 2}$

Step 5.1.5

Pull terms out from under the radical.

$d = \frac{30 \pm 30 \sqrt{7}}{2 \cdot 2}$

Step 5.2

Multiply $2$ by $2$ .

$d = \frac{30 \pm 30 \sqrt{7}}{4}$

Step 5.3

Simplify $\frac{30 \pm 30 \sqrt{7}}{4}$ .

$d = \frac{15 \pm 15 \sqrt{7}}{2}$

Step 5.4

Change the $\pm$ to $+$ .

$d = \frac{15 + 15 \sqrt{7}}{2}$

Step 6

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Tap for more steps...

Step 6.1

Simplify the numerator.

Tap for more steps...

Step 6.1.1

Raise $- 30$ to the power of $2$ .

$d = \frac{30 \pm \sqrt{900 - 4 \cdot 2 \cdot - 675}}{2 \cdot 2}$

Step 6.1.2

Multiply $- 4 \cdot 2 \cdot - 675$ .

Tap for more steps...

Step 6.1.2.1

Multiply $- 4$ by $2$ .

$d = \frac{30 \pm \sqrt{900 - 8 \cdot - 675}}{2 \cdot 2}$

Step 6.1.2.2

Multiply $- 8$ by $- 675$ .

$d = \frac{30 \pm \sqrt{900 + 5400}}{2 \cdot 2}$

Step 6.1.3

Add $900$ and $5400$ .

$d = \frac{30 \pm \sqrt{6300}}{2 \cdot 2}$

Step 6.1.4

Rewrite $6300$ as $30^{2} \cdot 7$ .

Tap for more steps...

Step 6.1.4.1

Factor $900$ out of $6300$ .

$d = \frac{30 \pm \sqrt{900 (7)}}{2 \cdot 2}$

Step 6.1.4.2

Rewrite $900$ as $30^{2}$ .

$d = \frac{30 \pm \sqrt{30^{2} \cdot 7}}{2 \cdot 2}$

Step 6.1.5

Pull terms out from under the radical.

$d = \frac{30 \pm 30 \sqrt{7}}{2 \cdot 2}$

Step 6.2

Multiply $2$ by $2$ .

$d = \frac{30 \pm 30 \sqrt{7}}{4}$

Step 6.3

Simplify $\frac{30 \pm 30 \sqrt{7}}{4}$ .

$d = \frac{15 \pm 15 \sqrt{7}}{2}$

Step 6.4

Change the $\pm$ to $-$ .

$d = \frac{15 - 15 \sqrt{7}}{2}$

Step 7

The final answer is the combination of both solutions.

$d = \frac{15 + 15 \sqrt{7}}{2}, \frac{15 - 15 \sqrt{7}}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$d = \frac{15 + 15 \sqrt{7}}{2}, \frac{15 - 15 \sqrt{7}}{2}$

Decimal Form:

$d = 27.34313483 \dots, - 12.34313483 \dots$