Algebra Examples

$h (x) = - \frac{16 x^{2}}{{(41)}^{2}} + \frac{71}{41} x$

Step 1

Simplify each term.

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Step 1.1

Raise $41$ to the power of $2$ .

$h (x) = - \frac{16 x^{2}}{1681} + \frac{71}{41} x$

Step 1.2

Cancel the common factor of $71$ and $41$ .

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Step 1.2.1

Rewrite $71$ as $1 (71)$ .

$h (x) = - \frac{16 x^{2}}{1681} + \frac{1 (71)}{41} x$

Step 1.2.2

Cancel the common factors.

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Step 1.2.2.1

Rewrite $41$ as $1 (41)$ .

$h (x) = - \frac{16 x^{2}}{1681} + \frac{1 \cdot 71}{1 \cdot 41} x$

Step 1.2.2.2

Cancel the common factor.

$h (x) = - \frac{16 x^{2}}{1681} + \frac{1 \cdot 71}{1 \cdot 41} x$

Step 1.2.2.3

Rewrite the expression.

$h (x) = - \frac{16 x^{2}}{1681} + \frac{71}{41} x$

Step 1.3

Combine $\frac{71}{41}$ and $x$ .

$h (x) = - \frac{16 x^{2}}{1681} + \frac{71 x}{41}$

Step 2

The maximum of a quadratic function occurs at $x = - \frac{b}{2 a}$ . If $a$ is negative, the maximum value of the function is $f (- \frac{b}{2 a})$ .

$f_{max}$ $x = a x^{2} + b x + c$ occurs at $x = - \frac{b}{2 a}$

Step 3

Find the value of $x = - \frac{b}{2 a}$ .

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Step 3.1

Substitute in the values of $a$ and $b$ .

$x = - \frac{1.\overline{73170}}{2 (- 0.00951814)}$

Step 3.2

Remove parentheses.

$x = - \frac{1.\overline{73170}}{2 (- 0.00951814)}$

Step 3.3

Simplify $- \frac{1.\overline{73170}}{2 (- 0.00951814)}$ .

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Step 3.3.1

Multiply $2$ by $- 0.00951814$ .

$x = - \frac{1.\overline{73170}}{- 0.01903628}$

Step 3.3.2

Divide $1.\overline{73170}$ by $- 0.01903628$ .

$x = - - 90.96875$

Step 3.3.3

Multiply $- 1$ by $- 90.96875$ .

$x = 90.96875$

Step 4

Evaluate $f (90.96875)$ .

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Step 4.1

Replace the variable $x$ with $90.96875$ in the expression.

$h (90.96875) = - \frac{16 {(90.96875)}^{2}}{1681} + \frac{71 (90.96875)}{41}$

Step 4.2

Simplify the result.

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Step 4.2.1

Multiply $71$ by $90.96875$ .

$h (90.96875) = - \frac{16 {(90.96875)}^{2}}{1681} + \frac{6458.78125}{41}$

Step 4.2.2

Simplify each term.

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Step 4.2.2.1

Raise $90.96875$ to the power of $2$ .

$h (90.96875) = - \frac{16 \cdot 8275.31347656}{1681} + \frac{6458.78125}{41}$

Step 4.2.2.2

Multiply $16$ by $8275.31347656$ .

$h (90.96875) = - \frac{132405.015625}{1681} + \frac{6458.78125}{41}$

Step 4.2.2.3

Divide $132405.015625$ by $1681$ .

$h (90.96875) = - 1 \cdot 78.765625 + \frac{6458.78125}{41}$

Step 4.2.2.4

Multiply $- 1$ by $78.765625$ .

$h (90.96875) = - 78.765625 + \frac{6458.78125}{41}$

Step 4.2.2.5

Divide $6458.78125$ by $41$ .

$h (90.96875) = - 78.765625 + 157.53125$

Step 4.2.3

Add $- 78.765625$ and $157.53125$ .

$h (90.96875) = 78.765625$

Step 4.2.4

The final answer is $78.765625$ .

$78.765625$

Step 5

Use the $x$ and $y$ values to find where the maximum occurs.

$(90.96875, 78.765625)$

Step 6