Pre-Algebra Examples

$1100 = \sqrt{9879 h + h^{2}}$

Step 1

Rewrite the equation as $\sqrt{9879 h + h^{2}} = 1100$ .

$\sqrt{9879 h + h^{2}} = 1100$

Step 2

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{9879 h + h^{2}}}^{2} = 1100^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9879 h + h^{2}}$ as ${(9879 h + h^{2})}^{\frac{1}{2}}$ .

${({(9879 h + h^{2})}^{\frac{1}{2}})}^{2} = 1100^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(9879 h + h^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(9879 h + h^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(9879 h + h^{2})}^{\frac{1}{2} \cdot 2} = 1100^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(9879 h + h^{2})}^{\frac{1}{2} \cdot 2} = 1100^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(9879 h + h^{2})}^{1} = 1100^{2}$

Step 3.2.1.2

Simplify.

$9879 h + h^{2} = 1100^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Raise $1100$ to the power of $2$ .

$9879 h + h^{2} = 1210000$

Step 4

Solve for $h$ .

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Step 4.1

Subtract $1210000$ from both sides of the equation.

$9879 h + h^{2} - 1210000 = 0$

Step 4.2

Factor the left side of the equation.

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Step 4.2.1

Let $u = h$ . Substitute $u$ for all occurrences of $h$ .

$9879 u + u^{2} - 1210000 = 0$

Step 4.2.2

Factor $9879 u + u^{2} - 1210000$ using the AC method.

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Step 4.2.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 1210000$ and whose sum is $9879$ .

$- 121, 10000$

Step 4.2.2.2

Write the factored form using these integers.

$(u - 121) (u + 10000) = 0$

Step 4.2.3

Replace all occurrences of $u$ with $h$ .

$(h - 121) (h + 10000) = 0$

Step 4.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$h - 121 = 0$

$h + 10000 = 0$

Step 4.4

Set $h - 121$ equal to $0$ and solve for $h$ .

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Step 4.4.1

Set $h - 121$ equal to $0$ .

$h - 121 = 0$

Step 4.4.2

Add $121$ to both sides of the equation.

$h = 121$

Step 4.5

Set $h + 10000$ equal to $0$ and solve for $h$ .

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Step 4.5.1

Set $h + 10000$ equal to $0$ .

$h + 10000 = 0$

Step 4.5.2

Subtract $10000$ from both sides of the equation.

$h = - 10000$

Step 4.6

The final solution is all the values that make $(h - 121) (h + 10000) = 0$ true.

$h = 121, - 10000$