Solve Using the Quadratic Formula x^2+(x+5)^2=25^2

$x^{2} + {(x + 5)}^{2} = 25^{2}$

Step 1

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

Simplify the right side.

Raise $25$ to the power of $2$ .

$x^{2} + {(x + 5)}^{2} = 625$

Subtract $625$ from both sides of the equation.

$x^{2} + {(x + 5)}^{2} - 625 = 0$

Simplify $x^{2} + {(x + 5)}^{2} - 625$ .

Simplify each term.

Rewrite ${(x + 5)}^{2}$ as $(x + 5) (x + 5)$ .

$x^{2} + (x + 5) (x + 5) - 625 = 0$

Expand $(x + 5) (x + 5)$ using the FOIL Method.

Apply the distributive property.

$x^{2} + x (x + 5) + 5 (x + 5) - 625 = 0$

Apply the distributive property.

$x^{2} + x \cdot x + x \cdot 5 + 5 (x + 5) - 625 = 0$

Apply the distributive property.

$x^{2} + x \cdot x + x \cdot 5 + 5 x + 5 \cdot 5 - 625 = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + x^{2} + x \cdot 5 + 5 x + 5 \cdot 5 - 625 = 0$

Move $5$ to the left of $x$ .

$x^{2} + x^{2} + 5 \cdot x + 5 x + 5 \cdot 5 - 625 = 0$

Multiply $5$ by $5$ .

$x^{2} + x^{2} + 5 x + 5 x + 25 - 625 = 0$

Add $5 x$ and $5 x$ .

$x^{2} + x^{2} + 10 x + 25 - 625 = 0$

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

$2 x^{2} + 10 x + 25 - 625 = 0$

Subtract $625$ from $25$ .

$2 x^{2} + 10 x - 600 = 0$

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 $a = 2$ , $b = 10$ , and $c = - 600$ into the quadratic formula and solve for $x$ .

$\frac{- 10 \pm \sqrt{10^{2} - 4 \cdot (2 \cdot - 600)}}{2 \cdot 2}$

Step 4

Simplify.

Simplify the numerator.

Raise $10$ to the power of $2$ .

$x = \frac{- 10 \pm \sqrt{100 - 4 \cdot 2 \cdot - 600}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot - 600$ .

Multiply $- 4$ by $2$ .

$x = \frac{- 10 \pm \sqrt{100 - 8 \cdot - 600}}{2 \cdot 2}$

Multiply $- 8$ by $- 600$ .

$x = \frac{- 10 \pm \sqrt{100 + 4800}}{2 \cdot 2}$

Add $100$ and $4800$ .

$x = \frac{- 10 \pm \sqrt{4900}}{2 \cdot 2}$

Rewrite $4900$ as $70^{2}$ .

$x = \frac{- 10 \pm \sqrt{70^{2}}}{2 \cdot 2}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- 10 \pm 70}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{- 10 \pm 70}{4}$

Simplify $\frac{- 10 \pm 70}{4}$ .

$x = \frac{- 5 \pm 35}{2}$

Step 5

The final answer is the combination of both solutions.

$x = 15, - 20$

Solve Using the Quadratic Formula x^2+(x+5)^2=25^2

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