Pre-Algebra Examples

$f (x) = - 50 x^{2} + 400 x - 750$

Step 1

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

Largest Degree: $2$

Leading Coefficient: $- 50$

Step 2

Simplify each term.

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Step 2.1

Cancel the common factor of $- 50$ .

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Step 2.1.1

Cancel the common factor.

$f (x) = \frac{- 50 x^{2}}{- 50} + \frac{400 x}{- 50} + \frac{- 750}{- 50}$

Step 2.1.2

Divide $x^{2}$ by $1$ .

$f (x) = x^{2} + \frac{400 x}{- 50} + \frac{- 750}{- 50}$

Step 2.2

Cancel the common factor of $400$ and $- 50$ .

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Step 2.2.1

Factor $50$ out of $400 x$ .

$f (x) = x^{2} + \frac{50 (8 x)}{- 50} + \frac{- 750}{- 50}$

Step 2.2.2

Move the negative one from the denominator of $\frac{8 x}{- 1}$ .

$f (x) = x^{2} - 1 \cdot (8 x) + \frac{- 750}{- 50}$

Step 2.3

Rewrite $- 1 \cdot (8 x)$ as $- (8 x)$ .

$f (x) = x^{2} - (8 x) + \frac{- 750}{- 50}$

Step 2.4

Multiply $8$ by $- 1$ .

$f (x) = x^{2} - 8 x + \frac{- 750}{- 50}$

Step 2.5

Divide $- 750$ by $- 50$ .

$f (x) = x^{2} - 8 x + 15$

Step 3

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

$- 8, 15$

Step 4

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 4.1

Arrange the terms in ascending order.

$b_{1} = | - 8 |, | 15 |$

Step 4.2

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

$b_{1} = | 15 |$

Step 4.3

The absolute value is the distance between a number and zero. The distance between $0$ and $15$ is $15$ .

$b_{1} = 15 + 1$

Step 4.4

Add $15$ and $1$ .

$b_{1} = 16$

Step 5

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 5.1

Simplify each term.

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Step 5.1.1

The absolute value is the distance between a number and zero. The distance between $- 8$ and $0$ is $8$ .

$b_{2} = 8 + | 15 |$

Step 5.1.2

The absolute value is the distance between a number and zero. The distance between $0$ and $15$ is $15$ .

$b_{2} = 8 + 15$

Step 5.2

Add $8$ and $15$ .

$b_{2} = 23$

Step 5.3

Arrange the terms in ascending order.

$b_{2} = 1, 23$

Step 5.4

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

$b_{2} = 23$

Step 6

Take the smaller bound option between $b_{1} = 16$ and $b_{2} = 23$ .

Smaller Bound: $16$

Step 7

Every real root on $f (x) = - 50 x^{2} + 400 x - 750$ lies between $- 16$ and $16$ .

$- 16$ and $16$