Pre-Algebra Examples

$- 7 \sqrt{13} - 8 \sqrt{x}$

Step 1

Write $- 7 \sqrt{13} - 8 \sqrt{x}$ as a function.

$f (x) = - 7 \sqrt{13} - 8 \sqrt{x}$

Step 2

Check the leading coefficient of the function. This number is the coefficient of the expression with the largest degree.

Largest Degree: $- 1$

Leading Coefficient: $- 8$

Step 3

Simplify each term.

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Step 3.1

Dividing two negative values results in a positive value.

$f (x) = \frac{7 \sqrt{13}}{8} + \frac{- 8 \sqrt{x}}{- 8}$

Step 3.2

Cancel the common factor of $- 8$ .

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Step 3.2.1

Cancel the common factor.

$f (x) = \frac{7 \sqrt{13}}{8} + \frac{- 8 \sqrt{x}}{- 8}$

Step 3.2.2

Divide $\sqrt{x}$ by $1$ .

$f (x) = \frac{7 \sqrt{13}}{8} + \sqrt{x}$

Step 4

Create a list of the coefficients of the function except the leading coefficient of $1$ .

$\frac{7}{8}$

Step 5

There will be two bound options, $b_{1}$ and $b_{2}$ , the smaller of which is the answer. To calculate the first bound option, find the absolute value of the largest coefficient from the list of coefficients. Then add $1$ .

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Step 5.1

Arrange the terms in ascending order.

$b_{1} = | \frac{7}{8} |$

Step 5.2

$\frac{7}{8}$ is approximately $0.875$ which is positive so remove the absolute value

$b_{1} = \frac{7}{8} + 1$

Step 5.3

Write $1$ as a fraction with a common denominator.

$b_{1} = \frac{7}{8} + \frac{8}{8}$

Step 5.4

Combine the numerators over the common denominator.

$b_{1} = \frac{7 + 8}{8}$

Step 5.5

Add $7$ and $8$ .

$b_{1} = \frac{15}{8}$

Step 6

To calculate the second bound option, sum the absolute values of the coefficients from the list of coefficients. If the sum is greater than $1$ , use that number. If not, use $1$ .

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Step 6.1

$\frac{7}{8}$ is approximately $0.875$ which is positive so remove the absolute value

$b_{2} = \frac{7}{8}$

Step 6.2

Arrange the terms in ascending order.

$b_{2} = \frac{7}{8}, 1$

Step 6.3

The maximum value is the largest value in the arranged data set.

$b_{2} = 1$

Step 7

Take the smaller bound option between $b_{1} = \frac{15}{8}$ and $b_{2} = 1$ .

Smaller Bound: $1$

Step 8

Every real root on $f (x) = - 7 \sqrt{13} - 8 \sqrt{x}$ lies between $- 1$ and $1$ .

$- 1$ and $1$