Precalculus Examples

$5 x^{3} - 3 x^{2} - 35 x + 21 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Factor out the greatest common factor from each group.

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Step 1.1.1

Group the first two terms and the last two terms.

$(5 x^{3} - 3 x^{2}) - 35 x + 21 = 0$

Step 1.1.2

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

$x^{2} (5 x - 3) - 7 (5 x - 3) = 0$

Step 1.2

Factor the polynomial by factoring out the greatest common factor, $5 x - 3$ .

$(5 x - 3) (x^{2} - 7) = 0$

Step 2

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

$5 x - 3 = 0$

$x^{2} - 7 = 0$

Step 3

Set $5 x - 3$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $5 x - 3$ equal to $0$ .

$5 x - 3 = 0$

Step 3.2

Solve $5 x - 3 = 0$ for $x$ .

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Step 3.2.1

Add $3$ to both sides of the equation.

$5 x = 3$

Step 3.2.2

Divide each term in $5 x = 3$ by $5$ and simplify.

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Step 3.2.2.1

Divide each term in $5 x = 3$ by $5$ .

$\frac{5 x}{5} = \frac{3}{5}$

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $5$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{3}{5}$

Step 3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{5}$

Step 4

Set $x^{2} - 7$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x^{2} - 7$ equal to $0$ .

$x^{2} - 7 = 0$

Step 4.2

Solve $x^{2} - 7 = 0$ for $x$ .

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Step 4.2.1

Add $7$ to both sides of the equation.

$x^{2} = 7$

Step 4.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{7}$

Step 4.2.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.2.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{7}$

Step 4.2.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{7}$

Step 4.2.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{7}, - \sqrt{7}$

Step 5

The final solution is all the values that make $(5 x - 3) (x^{2} - 7) = 0$ true.

$x = \frac{3}{5}, \sqrt{7}, - \sqrt{7}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3}{5}, \sqrt{7}, - \sqrt{7}$

Decimal Form:

$x = 0.6, 2.64575131 \dots, - 2.64575131 \dots$

Step 7