Precalculus Examples

$x^{2} - 10 x = 1575$

Step 1

To create a trinomial square on the left side of the equation, find a value that is equal to the square of half of $b$ .

${(\frac{b}{2})}^{2} = {(- 5)}^{2}$

Step 2

Add the term to each side of the equation.

$x^{2} - 10 x + {(- 5)}^{2} = 1575 + {(- 5)}^{2}$

Step 3

Simplify the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Raise $- 5$ to the power of $2$ .

$x^{2} - 10 x + 25 = 1575 + {(- 5)}^{2}$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $1575 + {(- 5)}^{2}$ .

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Step 3.2.1.1

Raise $- 5$ to the power of $2$ .

$x^{2} - 10 x + 25 = 1575 + 25$

Step 3.2.1.2

Add $1575$ and $25$ .

$x^{2} - 10 x + 25 = 1600$

Step 4

Factor the perfect trinomial square into ${(x - 5)}^{2}$ .

${(x - 5)}^{2} = 1600$

Step 5

Solve the equation for $x$ .

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Step 5.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x - 5 = \pm \sqrt{1600}$

Step 5.2

Simplify $\pm \sqrt{1600}$ .

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Step 5.2.1

Rewrite $1600$ as $40^{2}$ .

$x - 5 = \pm \sqrt{40^{2}}$

Step 5.2.2

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

$x - 5 = \pm 40$

Step 5.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x - 5 = 40$

Step 5.3.2

Move all terms not containing $x$ to the right side of the equation.

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Step 5.3.2.1

Add $5$ to both sides of the equation.

$x = 40 + 5$

Step 5.3.2.2

Add $40$ and $5$ .

$x = 45$

Step 5.3.3

Next, use the negative value of the $\pm$ to find the second solution.

$x - 5 = - 40$

Step 5.3.4

Move all terms not containing $x$ to the right side of the equation.

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Step 5.3.4.1

Add $5$ to both sides of the equation.

$x = - 40 + 5$

Step 5.3.4.2

Add $- 40$ and $5$ .

$x = - 35$

Step 5.3.5

The complete solution is the result of both the positive and negative portions of the solution.

$x = 45, - 35$