Precalculus Examples

$3 + \sqrt{4 a - 5} = 10$

Step 1

Move all terms not containing $\sqrt{4 a - 5}$ to the right side of the equation.

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Step 1.1

Subtract $3$ from both sides of the equation.

$\sqrt{4 a - 5} = 10 - 3$

Step 1.2

Subtract $3$ from $10$ .

$\sqrt{4 a - 5} = 7$

Step 2

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{4 a - 5}}^{2} = 7^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 a - 5}$ as ${(4 a - 5)}^{\frac{1}{2}}$ .

${({(4 a - 5)}^{\frac{1}{2}})}^{2} = 7^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(4 a - 5)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(4 a - 5)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(4 a - 5)}^{\frac{1}{2} \cdot 2} = 7^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(4 a - 5)}^{\frac{1}{2} \cdot 2} = 7^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(4 a - 5)}^{1} = 7^{2}$

Step 3.2.1.2

Simplify.

$4 a - 5 = 7^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Raise $7$ to the power of $2$ .

$4 a - 5 = 49$

Step 4

Solve for $a$ .

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Step 4.1

Move all terms not containing $a$ to the right side of the equation.

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Step 4.1.1

Add $5$ to both sides of the equation.

$4 a = 49 + 5$

Step 4.1.2

Add $49$ and $5$ .

$4 a = 54$

Step 4.2

Divide each term in $4 a = 54$ by $4$ and simplify.

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Step 4.2.1

Divide each term in $4 a = 54$ by $4$ .

$\frac{4 a}{4} = \frac{54}{4}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $4$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{4 a}{4} = \frac{54}{4}$

Step 4.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{54}{4}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Cancel the common factor of $54$ and $4$ .

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Step 4.2.3.1.1

Factor $2$ out of $54$ .

$a = \frac{2 (27)}{4}$

Step 4.2.3.1.2

Cancel the common factors.

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Step 4.2.3.1.2.1

Factor $2$ out of $4$ .

$a = \frac{2 \cdot 27}{2 \cdot 2}$

Step 4.2.3.1.2.2

Cancel the common factor.

$a = \frac{2 \cdot 27}{2 \cdot 2}$

Step 4.2.3.1.2.3

Rewrite the expression.

$a = \frac{27}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$a = \frac{27}{2}$

Decimal Form:

$a = 13.5$

Mixed Number Form:

$a = 13 \frac{1}{2}$