Solve Using the Quadratic Formula x^2+(x+21)^2=51^2

$x^{2} + {(x + 21)}^{2} = 51^{2}$

Step 1

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

Simplify the right side.

Raise $51$ to the power of $2$ .

$x^{2} + {(x + 21)}^{2} = 2601$

Subtract $2601$ from both sides of the equation.

$x^{2} + {(x + 21)}^{2} - 2601 = 0$

Simplify $x^{2} + {(x + 21)}^{2} - 2601$ .

Simplify each term.

Rewrite ${(x + 21)}^{2}$ as $(x + 21) (x + 21)$ .

$x^{2} + (x + 21) (x + 21) - 2601 = 0$

Expand $(x + 21) (x + 21)$ using the FOIL Method.

Apply the distributive property.

$x^{2} + x (x + 21) + 21 (x + 21) - 2601 = 0$

Apply the distributive property.

$x^{2} + x \cdot x + x \cdot 21 + 21 (x + 21) - 2601 = 0$

Apply the distributive property.

$x^{2} + x \cdot x + x \cdot 21 + 21 x + 21 \cdot 21 - 2601 = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + x^{2} + x \cdot 21 + 21 x + 21 \cdot 21 - 2601 = 0$

Move $21$ to the left of $x$ .

$x^{2} + x^{2} + 21 \cdot x + 21 x + 21 \cdot 21 - 2601 = 0$

Multiply $21$ by $21$ .

$x^{2} + x^{2} + 21 x + 21 x + 441 - 2601 = 0$

Add $21 x$ and $21 x$ .

$x^{2} + x^{2} + 42 x + 441 - 2601 = 0$

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

$2 x^{2} + 42 x + 441 - 2601 = 0$

Subtract $2601$ from $441$ .

$2 x^{2} + 42 x - 2160 = 0$

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 $a = 2$ , $b = 42$ , and $c = - 2160$ into the quadratic formula and solve for $x$ .

$\frac{- 42 \pm \sqrt{42^{2} - 4 \cdot (2 \cdot - 2160)}}{2 \cdot 2}$

Step 4

Simplify.

Simplify the numerator.

Raise $42$ to the power of $2$ .

$x = \frac{- 42 \pm \sqrt{1764 - 4 \cdot 2 \cdot - 2160}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot - 2160$ .

Multiply $- 4$ by $2$ .

$x = \frac{- 42 \pm \sqrt{1764 - 8 \cdot - 2160}}{2 \cdot 2}$

Multiply $- 8$ by $- 2160$ .

$x = \frac{- 42 \pm \sqrt{1764 + 17280}}{2 \cdot 2}$

Add $1764$ and $17280$ .

$x = \frac{- 42 \pm \sqrt{19044}}{2 \cdot 2}$

Rewrite $19044$ as $138^{2}$ .

$x = \frac{- 42 \pm \sqrt{138^{2}}}{2 \cdot 2}$

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

$x = \frac{- 42 \pm 138}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{- 42 \pm 138}{4}$

Simplify $\frac{- 42 \pm 138}{4}$ .

$x = \frac{- 21 \pm 69}{2}$

Step 5

The final answer is the combination of both solutions.

$x = 24, - 45$

Solve Using the Quadratic Formula x^2+(x+21)^2=51^2

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