Calculus Examples

$5 x (4 - x) - \frac{1}{2} - 6 \sqrt{4 - x} = 0$

Step 1

Simplify $5 x (4 - x) - \frac{1}{2} - 6 \sqrt{4 - x}$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$5 x \cdot 4 + 5 x (- x) - \frac{1}{2} - 6 \sqrt{4 - x} = 0$

Step 1.1.2

Multiply $4$ by $5$ .

$20 x + 5 x (- x) - \frac{1}{2} - 6 \sqrt{4 - x} = 0$

Step 1.1.3

Rewrite using the commutative property of multiplication.

$20 x + 5 \cdot - 1 x \cdot x - \frac{1}{2} - 6 \sqrt{4 - x} = 0$

Step 1.1.4

Simplify each term.

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Step 1.1.4.1

Multiply $x$ by $x$ by adding the exponents.

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Step 1.1.4.1.1

Move $x$ .

$20 x + 5 \cdot - 1 (x \cdot x) - \frac{1}{2} - 6 \sqrt{4 - x} = 0$

Step 1.1.4.1.2

Multiply $x$ by $x$ .

$20 x + 5 \cdot - 1 x^{2} - \frac{1}{2} - 6 \sqrt{4 - x} = 0$

Step 1.1.4.2

Multiply $5$ by $- 1$ .

$20 x - 5 x^{2} - \frac{1}{2} - 6 \sqrt{4 - x} = 0$

Step 1.2

To write $- 6 \sqrt{4 - x}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$20 x - 5 x^{2} - \frac{1}{2} - 6 \sqrt{4 - x} \cdot \frac{2}{2} = 0$

Step 1.3

Simplify terms.

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Step 1.3.1

Combine $- 6 \sqrt{4 - x}$ and $\frac{2}{2}$ .

$20 x - 5 x^{2} - \frac{1}{2} + \frac{- 6 \sqrt{4 - x} \cdot 2}{2} = 0$

Step 1.3.2

Combine the numerators over the common denominator.

$20 x - 5 x^{2} + \frac{- 1 - 6 \sqrt{4 - x} \cdot 2}{2} = 0$

Step 1.4

Simplify each term.

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Step 1.4.1

Simplify the numerator.

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Step 1.4.1.1

Factor $- 1$ out of $- 1 - 6 \sqrt{4 - x} \cdot 2$ .

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Step 1.4.1.1.1

Reorder $4$ and $- x$ .

$20 x - 5 x^{2} + \frac{- 1 - 6 \sqrt{- x + 4} \cdot 2}{2} = 0$

Step 1.4.1.1.2

Rewrite $- 1$ as $- 1 (1)$ .

$20 x - 5 x^{2} + \frac{- 1 (1) - 6 \sqrt{- x + 4} \cdot 2}{2} = 0$

Step 1.4.1.1.3

Factor $- 1$ out of $- 6 \sqrt{- x + 4} \cdot 2$ .

$20 x - 5 x^{2} + \frac{- 1 (1) - (6 \sqrt{- x + 4} \cdot 2)}{2} = 0$

Step 1.4.1.1.4

Factor $- 1$ out of $- 1 (1) - (6 \sqrt{- x + 4} \cdot 2)$ .

$20 x - 5 x^{2} + \frac{- 1 (1 + 6 \sqrt{- x + 4} \cdot 2)}{2} = 0$

Step 1.4.1.1.5

Rewrite $- 1 (1 + 6 \sqrt{- x + 4} \cdot 2)$ as $- (1 + 6 \sqrt{- x + 4} \cdot 2)$ .

$20 x - 5 x^{2} + \frac{- (1 + 6 \sqrt{- x + 4} \cdot 2)}{2} = 0$

Step 1.4.1.2

Multiply $2$ by $6$ .

$20 x - 5 x^{2} + \frac{- (1 + 12 \sqrt{- x + 4})}{2} = 0$

Step 1.4.2

Move the negative in front of the fraction.

$20 x - 5 x^{2} - \frac{1 + 12 \sqrt{- x + 4}}{2} = 0$

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx 0.68976564, 3.8559512$

Step 3