Solve Using the Quadratic Formula x^2+(x+140)^2=340^2

$x^{2} + {(x + 140)}^{2} = 340^{2}$

Step 1

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

Simplify the right side.

Raise $340$ to the power of $2$ .

$x^{2} + {(x + 140)}^{2} = 115600$

Subtract $115600$ from both sides of the equation.

$x^{2} + {(x + 140)}^{2} - 115600 = 0$

Simplify $x^{2} + {(x + 140)}^{2} - 115600$ .

Simplify each term.

Rewrite ${(x + 140)}^{2}$ as $(x + 140) (x + 140)$ .

$x^{2} + (x + 140) (x + 140) - 115600 = 0$

Expand $(x + 140) (x + 140)$ using the FOIL Method.

Apply the distributive property.

$x^{2} + x (x + 140) + 140 (x + 140) - 115600 = 0$

Apply the distributive property.

$x^{2} + x \cdot x + x \cdot 140 + 140 (x + 140) - 115600 = 0$

Apply the distributive property.

$x^{2} + x \cdot x + x \cdot 140 + 140 x + 140 \cdot 140 - 115600 = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + x^{2} + x \cdot 140 + 140 x + 140 \cdot 140 - 115600 = 0$

Move $140$ to the left of $x$ .

$x^{2} + x^{2} + 140 \cdot x + 140 x + 140 \cdot 140 - 115600 = 0$

Multiply $140$ by $140$ .

$x^{2} + x^{2} + 140 x + 140 x + 19600 - 115600 = 0$

Add $140 x$ and $140 x$ .

$x^{2} + x^{2} + 280 x + 19600 - 115600 = 0$

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

$2 x^{2} + 280 x + 19600 - 115600 = 0$

Subtract $115600$ from $19600$ .

$2 x^{2} + 280 x - 96000 = 0$

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 $a = 2$ , $b = 280$ , and $c = - 96000$ into the quadratic formula and solve for $x$ .

$\frac{- 280 \pm \sqrt{280^{2} - 4 \cdot (2 \cdot - 96000)}}{2 \cdot 2}$

Step 4

Simplify.

Simplify the numerator.

Raise $280$ to the power of $2$ .

$x = \frac{- 280 \pm \sqrt{78400 - 4 \cdot 2 \cdot - 96000}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot - 96000$ .

Multiply $- 4$ by $2$ .

$x = \frac{- 280 \pm \sqrt{78400 - 8 \cdot - 96000}}{2 \cdot 2}$

Multiply $- 8$ by $- 96000$ .

$x = \frac{- 280 \pm \sqrt{78400 + 768000}}{2 \cdot 2}$

Add $78400$ and $768000$ .

$x = \frac{- 280 \pm \sqrt{846400}}{2 \cdot 2}$

Rewrite $846400$ as $920^{2}$ .

$x = \frac{- 280 \pm \sqrt{920^{2}}}{2 \cdot 2}$

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

$x = \frac{- 280 \pm 920}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{- 280 \pm 920}{4}$

Simplify $\frac{- 280 \pm 920}{4}$ .

$x = - 70 \pm 230$

Step 5

The final answer is the combination of both solutions.

$x = 160, - 300$

Solve Using the Quadratic Formula x^2+(x+140)^2=340^2

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