Basic Math Examples

$2 z + 3 z^{\frac{1}{2}} - 2 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Rewrite $z$ as ${(z^{\frac{1}{2}})}^{2}$ .

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

Step 1.2

Let $u = z^{\frac{1}{2}}$ . Substitute $u$ for all occurrences of $z^{\frac{1}{2}}$ .

$2 u^{2} + 3 u - 2 = 0$

Step 1.3

Factor by grouping.

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Step 1.3.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 2 = - 4$ and whose sum is $b = 3$ .

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Step 1.3.1.1

Factor $3$ out of $3 u$ .

$2 u^{2} + 3 (u) - 2 = 0$

Step 1.3.1.2

Rewrite $3$ as $- 1$ plus $4$

$2 u^{2} + (- 1 + 4) u - 2 = 0$

Step 1.3.1.3

Apply the distributive property.

$2 u^{2} - 1 u + 4 u - 2 = 0$

Step 1.3.2

Factor out the greatest common factor from each group.

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Step 1.3.2.1

Group the first two terms and the last two terms.

$(2 u^{2} - 1 u) + 4 u - 2 = 0$

Step 1.3.2.2

Factor out the greatest common factor (GCF) from each group.

$u (2 u - 1) + 2 (2 u - 1) = 0$

Step 1.3.3

Factor the polynomial by factoring out the greatest common factor, $2 u - 1$ .

$(2 u - 1) (u + 2) = 0$

Step 1.4

Replace all occurrences of $u$ with $z^{\frac{1}{2}}$ .

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

Step 2

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

$2 z^{\frac{1}{2}} - 1 = 0$

$z^{\frac{1}{2}} + 2 = 0$

Step 3

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

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Step 3.1

Set $2 z^{\frac{1}{2}} - 1$ equal to $0$ .

$2 z^{\frac{1}{2}} - 1 = 0$

Step 3.2

Solve $2 z^{\frac{1}{2}} - 1 = 0$ for $z$ .

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Step 3.2.1

Add $1$ to both sides of the equation.

$2 z^{\frac{1}{2}} = 1$

Step 3.2.2

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${(2 z^{\frac{1}{2}})}^{2} = 1^{2}$

Step 3.2.3

Simplify the exponent.

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Step 3.2.3.1

Simplify the left side.

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Step 3.2.3.1.1

Simplify ${(2 z^{\frac{1}{2}})}^{2}$ .

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Step 3.2.3.1.1.1

Apply the product rule to $2 z^{\frac{1}{2}}$ .

$2^{2} {(z^{\frac{1}{2}})}^{2} = 1^{2}$

Step 3.2.3.1.1.2

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

$4 {(z^{\frac{1}{2}})}^{2} = 1^{2}$

Step 3.2.3.1.1.3

Multiply the exponents in ${(z^{\frac{1}{2}})}^{2}$ .

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Step 3.2.3.1.1.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$4 z^{\frac{1}{2} \cdot 2} = 1^{2}$

Step 3.2.3.1.1.3.2

Cancel the common factor of $2$ .

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Step 3.2.3.1.1.3.2.1

Cancel the common factor.

$4 z^{\frac{1}{2} \cdot 2} = 1^{2}$

Step 3.2.3.1.1.3.2.2

Rewrite the expression.

$4 z^{1} = 1^{2}$

Step 3.2.3.1.1.4

Simplify.

$4 z = 1^{2}$

Step 3.2.3.2

Simplify the right side.

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Step 3.2.3.2.1

One to any power is one.

$4 z = 1$

Step 3.2.4

Divide each term in $4 z = 1$ by $4$ and simplify.

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Step 3.2.4.1

Divide each term in $4 z = 1$ by $4$ .

$\frac{4 z}{4} = \frac{1}{4}$

Step 3.2.4.2

Simplify the left side.

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Step 3.2.4.2.1

Cancel the common factor of $4$ .

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Step 3.2.4.2.1.1

Cancel the common factor.

$\frac{4 z}{4} = \frac{1}{4}$

Step 3.2.4.2.1.2

Divide $z$ by $1$ .

$z = \frac{1}{4}$

Step 4

Set $z^{\frac{1}{2}} + 2$ equal to $0$ and solve for $z$ .

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Step 4.1

Set $z^{\frac{1}{2}} + 2$ equal to $0$ .

$z^{\frac{1}{2}} + 2 = 0$

Step 4.2

Solve $z^{\frac{1}{2}} + 2 = 0$ for $z$ .

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Step 4.2.1

Subtract $2$ from both sides of the equation.

$z^{\frac{1}{2}} = - 2$

Step 4.2.2

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

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

Step 4.2.3

Simplify the exponent.

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Step 4.2.3.1

Simplify the left side.

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Step 4.2.3.1.1

Simplify ${(z^{\frac{1}{2}})}^{2}$ .

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Step 4.2.3.1.1.1

Multiply the exponents in ${(z^{\frac{1}{2}})}^{2}$ .

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Step 4.2.3.1.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$z^{\frac{1}{2} \cdot 2} = {(- 2)}^{2}$

Step 4.2.3.1.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.3.1.1.1.2.1

Cancel the common factor.

$z^{\frac{1}{2} \cdot 2} = {(- 2)}^{2}$

Step 4.2.3.1.1.1.2.2

Rewrite the expression.

$z^{1} = {(- 2)}^{2}$

Step 4.2.3.1.1.2

Simplify.

$z = {(- 2)}^{2}$

Step 4.2.3.2

Simplify the right side.

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Step 4.2.3.2.1

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

$z = 4$

Step 5

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

$z = \frac{1}{4}, 4$

Step 6

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

$z = \frac{1}{4}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$z = \frac{1}{4}$

Decimal Form:

$z = 0.25$