Solve Using the Quadratic Formula x^2+(x+7)^2=35^2

$x^{2} + {(x + 7)}^{2} = 35^{2}$

Step 1

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

Simplify the right side.

Raise $35$ to the power of $2$ .

$x^{2} + {(x + 7)}^{2} = 1225$

Subtract $1225$ from both sides of the equation.

$x^{2} + {(x + 7)}^{2} - 1225 = 0$

Simplify $x^{2} + {(x + 7)}^{2} - 1225$ .

Simplify each term.

Rewrite ${(x + 7)}^{2}$ as $(x + 7) (x + 7)$ .

$x^{2} + (x + 7) (x + 7) - 1225 = 0$

Expand $(x + 7) (x + 7)$ using the FOIL Method.

Apply the distributive property.

$x^{2} + x (x + 7) + 7 (x + 7) - 1225 = 0$

Apply the distributive property.

$x^{2} + x \cdot x + x \cdot 7 + 7 (x + 7) - 1225 = 0$

Apply the distributive property.

$x^{2} + x \cdot x + x \cdot 7 + 7 x + 7 \cdot 7 - 1225 = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + x^{2} + x \cdot 7 + 7 x + 7 \cdot 7 - 1225 = 0$

Move $7$ to the left of $x$ .

$x^{2} + x^{2} + 7 \cdot x + 7 x + 7 \cdot 7 - 1225 = 0$

Multiply $7$ by $7$ .

$x^{2} + x^{2} + 7 x + 7 x + 49 - 1225 = 0$

Add $7 x$ and $7 x$ .

$x^{2} + x^{2} + 14 x + 49 - 1225 = 0$

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

$2 x^{2} + 14 x + 49 - 1225 = 0$

Subtract $1225$ from $49$ .

$2 x^{2} + 14 x - 1176 = 0$

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 $a = 2$ , $b = 14$ , and $c = - 1176$ into the quadratic formula and solve for $x$ .

$\frac{- 14 \pm \sqrt{14^{2} - 4 \cdot (2 \cdot - 1176)}}{2 \cdot 2}$

Step 4

Simplify.

Simplify the numerator.

Raise $14$ to the power of $2$ .

$x = \frac{- 14 \pm \sqrt{196 - 4 \cdot 2 \cdot - 1176}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot - 1176$ .

Multiply $- 4$ by $2$ .

$x = \frac{- 14 \pm \sqrt{196 - 8 \cdot - 1176}}{2 \cdot 2}$

Multiply $- 8$ by $- 1176$ .

$x = \frac{- 14 \pm \sqrt{196 + 9408}}{2 \cdot 2}$

Add $196$ and $9408$ .

$x = \frac{- 14 \pm \sqrt{9604}}{2 \cdot 2}$

Rewrite $9604$ as $98^{2}$ .

$x = \frac{- 14 \pm \sqrt{98^{2}}}{2 \cdot 2}$

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

$x = \frac{- 14 \pm 98}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{- 14 \pm 98}{4}$

Simplify $\frac{- 14 \pm 98}{4}$ .

$x = \frac{- 7 \pm 49}{2}$

Step 5

The final answer is the combination of both solutions.

$x = 21, - 28$

Solve Using the Quadratic Formula x^2+(x+7)^2=35^2

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