Precalculus Examples

$6 x - 7 \sqrt{x} - 3 = 0$

Step 1

Move all terms not containing $- 7 \sqrt{x}$ to the right side of the equation.

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Step 1.1

Subtract $6 x$ from both sides of the equation.

$- 7 \sqrt{x} - 3 = - 6 x$

Step 1.2

Add $3$ to both sides of the equation.

$- 7 \sqrt{x} = - 6 x + 3$

Step 2

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

${(- 7 \sqrt{x})}^{2} = {(- 6 x + 3)}^{2}$

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 $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

${(- 7 x^{\frac{1}{2}})}^{2} = {(- 6 x + 3)}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${(- 7 x^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Apply the product rule to $- 7 x^{\frac{1}{2}}$ .

${(- 7)}^{2} {(x^{\frac{1}{2}})}^{2} = {(- 6 x + 3)}^{2}$

Step 3.2.1.2

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

$49 {(x^{\frac{1}{2}})}^{2} = {(- 6 x + 3)}^{2}$

Step 3.2.1.3

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

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Step 3.2.1.3.1

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

$49 x^{\frac{1}{2} \cdot 2} = {(- 6 x + 3)}^{2}$

Step 3.2.1.3.2

Cancel the common factor of $2$ .

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Step 3.2.1.3.2.1

Cancel the common factor.

$49 x^{\frac{1}{2} \cdot 2} = {(- 6 x + 3)}^{2}$

Step 3.2.1.3.2.2

Rewrite the expression.

$49 x^{1} = {(- 6 x + 3)}^{2}$

Step 3.2.1.4

Simplify.

$49 x = {(- 6 x + 3)}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(- 6 x + 3)}^{2}$ .

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Step 3.3.1.1

Rewrite ${(- 6 x + 3)}^{2}$ as $(- 6 x + 3) (- 6 x + 3)$ .

$49 x = (- 6 x + 3) (- 6 x + 3)$

Step 3.3.1.2

Expand $(- 6 x + 3) (- 6 x + 3)$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$49 x = - 6 x (- 6 x + 3) + 3 (- 6 x + 3)$

Step 3.3.1.2.2

Apply the distributive property.

$49 x = - 6 x (- 6 x) - 6 x \cdot 3 + 3 (- 6 x + 3)$

Step 3.3.1.2.3

Apply the distributive property.

$49 x = - 6 x (- 6 x) - 6 x \cdot 3 + 3 (- 6 x) + 3 \cdot 3$

Step 3.3.1.3

Simplify and combine like terms.

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Step 3.3.1.3.1

Simplify each term.

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Step 3.3.1.3.1.1

Rewrite using the commutative property of multiplication.

$49 x = - 6 \cdot - 6 x \cdot x - 6 x \cdot 3 + 3 (- 6 x) + 3 \cdot 3$

Step 3.3.1.3.1.2

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

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Step 3.3.1.3.1.2.1

Move $x$ .

$49 x = - 6 \cdot - 6 (x \cdot x) - 6 x \cdot 3 + 3 (- 6 x) + 3 \cdot 3$

Step 3.3.1.3.1.2.2

Multiply $x$ by $x$ .

$49 x = - 6 \cdot - 6 x^{2} - 6 x \cdot 3 + 3 (- 6 x) + 3 \cdot 3$

Step 3.3.1.3.1.3

Multiply $- 6$ by $- 6$ .

$49 x = 36 x^{2} - 6 x \cdot 3 + 3 (- 6 x) + 3 \cdot 3$

Step 3.3.1.3.1.4

Multiply $3$ by $- 6$ .

$49 x = 36 x^{2} - 18 x + 3 (- 6 x) + 3 \cdot 3$

Step 3.3.1.3.1.5

Multiply $- 6$ by $3$ .

$49 x = 36 x^{2} - 18 x - 18 x + 3 \cdot 3$

Step 3.3.1.3.1.6

Multiply $3$ by $3$ .

$49 x = 36 x^{2} - 18 x - 18 x + 9$

Step 3.3.1.3.2

Subtract $18 x$ from $- 18 x$ .

$49 x = 36 x^{2} - 36 x + 9$

Step 4

Solve for $x$ .

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Step 4.1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$36 x^{2} - 36 x + 9 = 49 x$

Step 4.2

Move all terms containing $x$ to the left side of the equation.

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Step 4.2.1

Subtract $49 x$ from both sides of the equation.

$36 x^{2} - 36 x + 9 - 49 x = 0$

Step 4.2.2

Subtract $49 x$ from $- 36 x$ .

$36 x^{2} - 85 x + 9 = 0$

Step 4.3

Factor by grouping.

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Step 4.3.1

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 $a \cdot c = 36 \cdot 9 = 324$ and whose sum is $b = - 85$ .

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Step 4.3.1.1

Factor $- 85$ out of $- 85 x$ .

$36 x^{2} - 85 x + 9 = 0$

Step 4.3.1.2

Rewrite $- 85$ as $- 4$ plus $- 81$

$36 x^{2} + (- 4 - 81) x + 9 = 0$

Step 4.3.1.3

Apply the distributive property.

$36 x^{2} - 4 x - 81 x + 9 = 0$

Step 4.3.2

Factor out the greatest common factor from each group.

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Step 4.3.2.1

Group the first two terms and the last two terms.

$(36 x^{2} - 4 x) - 81 x + 9 = 0$

Step 4.3.2.2

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

$4 x (9 x - 1) - 9 (9 x - 1) = 0$

Step 4.3.3

Factor the polynomial by factoring out the greatest common factor, $9 x - 1$ .

$(9 x - 1) (4 x - 9) = 0$

Step 4.4

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

$9 x - 1 = 0$

$4 x - 9 = 0$

Step 4.5

Set $9 x - 1$ equal to $0$ and solve for $x$ .

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Step 4.5.1

Set $9 x - 1$ equal to $0$ .

$9 x - 1 = 0$

Step 4.5.2

Solve $9 x - 1 = 0$ for $x$ .

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Step 4.5.2.1

Add $1$ to both sides of the equation.

$9 x = 1$

Step 4.5.2.2

Divide each term in $9 x = 1$ by $9$ and simplify.

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Step 4.5.2.2.1

Divide each term in $9 x = 1$ by $9$ .

$\frac{9 x}{9} = \frac{1}{9}$

Step 4.5.2.2.2

Simplify the left side.

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Step 4.5.2.2.2.1

Cancel the common factor of $9$ .

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Step 4.5.2.2.2.1.1

Cancel the common factor.

$\frac{9 x}{9} = \frac{1}{9}$

Step 4.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{9}$

Step 4.6

Set $4 x - 9$ equal to $0$ and solve for $x$ .

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Step 4.6.1

Set $4 x - 9$ equal to $0$ .

$4 x - 9 = 0$

Step 4.6.2

Solve $4 x - 9 = 0$ for $x$ .

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Step 4.6.2.1

Add $9$ to both sides of the equation.

$4 x = 9$

Step 4.6.2.2

Divide each term in $4 x = 9$ by $4$ and simplify.

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Step 4.6.2.2.1

Divide each term in $4 x = 9$ by $4$ .

$\frac{4 x}{4} = \frac{9}{4}$

Step 4.6.2.2.2

Simplify the left side.

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Step 4.6.2.2.2.1

Cancel the common factor of $4$ .

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Step 4.6.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{9}{4}$

Step 4.6.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{9}{4}$

Step 4.7

The final solution is all the values that make $(9 x - 1) (4 x - 9) = 0$ true.

$x = \frac{1}{9}, \frac{9}{4}$

Step 5

Exclude the solutions that do not make $6 x - 7 \sqrt{x} - 3 = 0$ true.

$x = \frac{9}{4}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{9}{4}$

Decimal Form:

$x = 2.25$

Mixed Number Form:

$x = 2 \frac{1}{4}$