Pre-Algebra Examples

$y = - 5 x^{2} + 100 x + 50$

Step 1

Write $y = - 5 x^{2} + 100 x + 50$ as a function.

$f (x) = - 5 x^{2} + 100 x + 50$

Step 2

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

Largest Degree: $2$

Leading Coefficient: $- 5$

Step 3

Simplify each term.

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Step 3.1

Cancel the common factor of $- 5$ .

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Step 3.1.1

Cancel the common factor.

$f (x) = \frac{- 5 x^{2}}{- 5} + \frac{100 x}{- 5} + \frac{50}{- 5}$

Step 3.1.2

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

$f (x) = x^{2} + \frac{100 x}{- 5} + \frac{50}{- 5}$

Step 3.2

Cancel the common factor of $100$ and $- 5$ .

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Step 3.2.1

Factor $5$ out of $100 x$ .

$f (x) = x^{2} + \frac{5 (20 x)}{- 5} + \frac{50}{- 5}$

Step 3.2.2

Move the negative one from the denominator of $\frac{20 x}{- 1}$ .

$f (x) = x^{2} - 1 \cdot (20 x) + \frac{50}{- 5}$

Step 3.3

Rewrite $- 1 \cdot (20 x)$ as $- (20 x)$ .

$f (x) = x^{2} - (20 x) + \frac{50}{- 5}$

Step 3.4

Multiply $20$ by $- 1$ .

$f (x) = x^{2} - 20 x + \frac{50}{- 5}$

Step 3.5

Divide $50$ by $- 5$ .

$f (x) = x^{2} - 20 x - 10$

Step 4

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

$- 20, - 10$

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.

$b_{1} = | - 10 |, | - 20 |$

Step 5.2

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

$b_{1} = | - 20 |$

Step 5.3

The absolute value is the distance between a number and zero. The distance between $- 20$ and $0$ is $20$ .

$b_{1} = 20 + 1$

Step 5.4

Add $20$ and $1$ .

$b_{1} = 21$

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

Simplify each term.

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Step 6.1.1

The absolute value is the distance between a number and zero. The distance between $- 20$ and $0$ is $20$ .

$b_{2} = 20 + | - 10 |$

Step 6.1.2

The absolute value is the distance between a number and zero. The distance between $- 10$ and $0$ is $10$ .

$b_{2} = 20 + 10$

Step 6.2

Add $20$ and $10$ .

$b_{2} = 30$

Step 6.3

Arrange the terms in ascending order.

$b_{2} = 1, 30$

Step 6.4

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

$b_{2} = 30$

Step 7

Take the smaller bound option between $b_{1} = 21$ and $b_{2} = 30$ .

Smaller Bound: $21$

Step 8

Every real root on $f (x) = - 5 x^{2} + 100 x + 50$ lies between $- 21$ and $21$ .

$- 21$ and $21$