Precalculus Examples

$3 x {(x - 1)}^{\frac{1}{2}} + 2 {(x - 1)}^{\frac{3}{2}} = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Factor ${(x - 1)}^{\frac{1}{2}}$ out of $3 x {(x - 1)}^{\frac{1}{2}} + 2 {(x - 1)}^{\frac{3}{2}}$ .

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Step 1.1.1

Factor ${(x - 1)}^{\frac{1}{2}}$ out of $3 x {(x - 1)}^{\frac{1}{2}}$ .

${(x - 1)}^{\frac{1}{2}} (3 x) + 2 {(x - 1)}^{\frac{3}{2}} = 0$

Step 1.1.2

Factor ${(x - 1)}^{\frac{1}{2}}$ out of $2 {(x - 1)}^{\frac{3}{2}}$ .

${(x - 1)}^{\frac{1}{2}} (3 x) + {(x - 1)}^{\frac{1}{2}} (2 {(x - 1)}^{\frac{2}{2}}) = 0$

Step 1.1.3

Factor ${(x - 1)}^{\frac{1}{2}}$ out of ${(x - 1)}^{\frac{1}{2}} (3 x) + {(x - 1)}^{\frac{1}{2}} (2 {(x - 1)}^{\frac{2}{2}})$ .

${(x - 1)}^{\frac{1}{2}} (3 x + 2 {(x - 1)}^{\frac{2}{2}}) = 0$

Step 1.2

Divide $2$ by $2$ .

${(x - 1)}^{\frac{1}{2}} (3 x + 2 (x - 1)) = 0$

Step 1.3

Simplify.

${(x - 1)}^{\frac{1}{2}} (3 x + 2 (x - 1)) = 0$

Step 1.4

Apply the distributive property.

${(x - 1)}^{\frac{1}{2}} (3 x + 2 x + 2 \cdot - 1) = 0$

Step 1.5

Multiply $2$ by $- 1$ .

${(x - 1)}^{\frac{1}{2}} (3 x + 2 x - 2) = 0$

Step 1.6

Add $3 x$ and $2 x$ .

${(x - 1)}^{\frac{1}{2}} (5 x - 2) = 0$

Step 2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

${(x - 1)}^{\frac{1}{2}} = 0$

$5 x - 2 = 0$

Step 3

Set ${(x - 1)}^{\frac{1}{2}}$ equal to $0$ and solve for $x$ .

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Step 3.1

Set ${(x - 1)}^{\frac{1}{2}}$ equal to $0$ .

${(x - 1)}^{\frac{1}{2}} = 0$

Step 3.2

Solve ${(x - 1)}^{\frac{1}{2}} = 0$ for $x$ .

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Step 3.2.1

Set the $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 3.2.2

Add $1$ to both sides of the equation.

$x = 1$

Step 4

Set $5 x - 2$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $5 x - 2$ equal to $0$ .

$5 x - 2 = 0$

Step 4.2

Solve $5 x - 2 = 0$ for $x$ .

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Step 4.2.1

Add $2$ to both sides of the equation.

$5 x = 2$

Step 4.2.2

Divide each term in $5 x = 2$ by $5$ and simplify.

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Step 4.2.2.1

Divide each term in $5 x = 2$ by $5$ .

$\frac{5 x}{5} = \frac{2}{5}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of $5$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{2}{5}$

Step 4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{2}{5}$

Step 5

The final solution is all the values that make ${(x - 1)}^{\frac{1}{2}} (5 x - 2) = 0$ true.

$x = 1, \frac{2}{5}$

Step 6

Exclude the solutions that do not make $3 x {(x - 1)}^{\frac{1}{2}} + 2 {(x - 1)}^{\frac{3}{2}} = 0$ true.

$x = 1$