Pre-Algebra Examples

$f (x) = - 4 x^{2} + 10 x - 3$

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: $- 4$

Step 2

Simplify each term.

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Step 2.1

Cancel the common factor of $- 4$ .

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Step 2.1.1

Cancel the common factor.

$f (x) = \frac{- 4 x^{2}}{- 4} + \frac{10 x}{- 4} + \frac{- 3}{- 4}$

Step 2.1.2

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

$f (x) = x^{2} + \frac{10 x}{- 4} + \frac{- 3}{- 4}$

Step 2.2

Cancel the common factor of $10$ and $- 4$ .

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Step 2.2.1

Factor $2$ out of $10 x$ .

$f (x) = x^{2} + \frac{2 (5 x)}{- 4} + \frac{- 3}{- 4}$

Step 2.2.2

Cancel the common factors.

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Step 2.2.2.1

Factor $2$ out of $- 4$ .

$f (x) = x^{2} + \frac{2 (5 x)}{2 (- 2)} + \frac{- 3}{- 4}$

Step 2.2.2.2

Cancel the common factor.

$f (x) = x^{2} + \frac{2 (5 x)}{2 \cdot - 2} + \frac{- 3}{- 4}$

Step 2.2.2.3

Rewrite the expression.

$f (x) = x^{2} + \frac{5 x}{- 2} + \frac{- 3}{- 4}$

Step 2.3

Move the negative in front of the fraction.

$f (x) = x^{2} - \frac{5 x}{2} + \frac{- 3}{- 4}$

Step 2.4

Dividing two negative values results in a positive value.

$f (x) = x^{2} - \frac{5 x}{2} + \frac{3}{4}$

Step 3

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

$- \frac{5}{2}, \frac{3}{4}$

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.

$b_{1} = | \frac{3}{4} |, | - \frac{5}{2} |$

Step 4.2

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

$b_{1} = | - \frac{5}{2} |$

Step 4.3

$- \frac{5}{2}$ is approximately $- 2.5$ which is negative so negate $- \frac{5}{2}$ and remove the absolute value

$b_{1} = \frac{5}{2} + 1$

Step 4.4

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

$b_{1} = \frac{5}{2} + \frac{2}{2}$

Step 4.5

Combine the numerators over the common denominator.

$b_{1} = \frac{5 + 2}{2}$

Step 4.6

Add $5$ and $2$ .

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

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

$- \frac{5}{2}$ is approximately $- 2.5$ which is negative so negate $- \frac{5}{2}$ and remove the absolute value

$b_{2} = \frac{5}{2} + | \frac{3}{4} |$

Step 5.1.2

$\frac{3}{4}$ is approximately $0.75$ which is positive so remove the absolute value

$b_{2} = \frac{5}{2} + \frac{3}{4}$

Step 5.2

To write $\frac{5}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$b_{2} = \frac{5}{2} \cdot \frac{2}{2} + \frac{3}{4}$

Step 5.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.3.1

Multiply $\frac{5}{2}$ by $\frac{2}{2}$ .

$b_{2} = \frac{5 \cdot 2}{2 \cdot 2} + \frac{3}{4}$

Step 5.3.2

Multiply $2$ by $2$ .

$b_{2} = \frac{5 \cdot 2}{4} + \frac{3}{4}$

Step 5.4

Combine the numerators over the common denominator.

$b_{2} = \frac{5 \cdot 2 + 3}{4}$

Step 5.5

Simplify the numerator.

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Step 5.5.1

Multiply $5$ by $2$ .

$b_{2} = \frac{10 + 3}{4}$

Step 5.5.2

Add $10$ and $3$ .

$b_{2} = \frac{13}{4}$

Step 5.6

Arrange the terms in ascending order.

$b_{2} = 1, \frac{13}{4}$

Step 5.7

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

$b_{2} = \frac{13}{4}$

Step 6

Take the smaller bound option between $b_{1} = \frac{7}{2}$ and $b_{2} = \frac{13}{4}$ .

Smaller Bound: $\frac{13}{4}$

Step 7

Every real root on $f (x) = - 4 x^{2} + 10 x - 3$ lies between $- \frac{13}{4}$ and $\frac{13}{4}$ .

$- \frac{13}{4}$ and $\frac{13}{4}$