Algebra Examples

$f (x) = 60 x^{2} - 57 x - 18$

Step 1

Set $60 x^{2} - 57 x - 18$ equal to $0$ .

$60 x^{2} - 57 x - 18 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor the left side of the equation.

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Step 2.1.1

Factor $3$ out of $60 x^{2} - 57 x - 18$ .

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Step 2.1.1.1

Factor $3$ out of $60 x^{2}$ .

$3 (20 x^{2}) - 57 x - 18 = 0$

Step 2.1.1.2

Factor $3$ out of $- 57 x$ .

$3 (20 x^{2}) + 3 (- 19 x) - 18 = 0$

Step 2.1.1.3

Factor $3$ out of $- 18$ .

$3 (20 x^{2}) + 3 (- 19 x) + 3 (- 6) = 0$

Step 2.1.1.4

Factor $3$ out of $3 (20 x^{2}) + 3 (- 19 x)$ .

$3 (20 x^{2} - 19 x) + 3 (- 6) = 0$

Step 2.1.1.5

Factor $3$ out of $3 (20 x^{2} - 19 x) + 3 (- 6)$ .

$3 (20 x^{2} - 19 x - 6) = 0$

Step 2.1.2

Factor.

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Step 2.1.2.1

Factor by grouping.

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Step 2.1.2.1.1

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 $a \cdot c = 20 \cdot - 6 = - 120$ and whose sum is $b = - 19$ .

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Step 2.1.2.1.1.1

Factor $- 19$ out of $- 19 x$ .

$3 (20 x^{2} - 19 x - 6) = 0$

Step 2.1.2.1.1.2

Rewrite $- 19$ as $5$ plus $- 24$

$3 (20 x^{2} + (5 - 24) x - 6) = 0$

Step 2.1.2.1.1.3

Apply the distributive property.

$3 (20 x^{2} + 5 x - 24 x - 6) = 0$

Step 2.1.2.1.2

Factor out the greatest common factor from each group.

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Step 2.1.2.1.2.1

Group the first two terms and the last two terms.

$3 ((20 x^{2} + 5 x) - 24 x - 6) = 0$

Step 2.1.2.1.2.2

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

$3 (5 x (4 x + 1) - 6 (4 x + 1)) = 0$

Step 2.1.2.1.3

Factor the polynomial by factoring out the greatest common factor, $4 x + 1$ .

$3 ((4 x + 1) (5 x - 6)) = 0$

Step 2.1.2.2

Remove unnecessary parentheses.

$3 (4 x + 1) (5 x - 6) = 0$

Step 2.2

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

$4 x + 1 = 0$

$5 x - 6 = 0$

Step 2.3

Set $4 x + 1$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set $4 x + 1$ equal to $0$ .

$4 x + 1 = 0$

Step 2.3.2

Solve $4 x + 1 = 0$ for $x$ .

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Step 2.3.2.1

Subtract $1$ from both sides of the equation.

$4 x = - 1$

Step 2.3.2.2

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

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Step 2.3.2.2.1

Divide each term in $4 x = - 1$ by $4$ .

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

Step 2.3.2.2.2

Simplify the left side.

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Step 2.3.2.2.2.1

Cancel the common factor of $4$ .

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Step 2.3.2.2.2.1.1

Cancel the common factor.

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

Step 2.3.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 2.3.2.2.3

Simplify the right side.

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Step 2.3.2.2.3.1

Move the negative in front of the fraction.

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

Step 2.4

Set $5 x - 6$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $5 x - 6$ equal to $0$ .

$5 x - 6 = 0$

Step 2.4.2

Solve $5 x - 6 = 0$ for $x$ .

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Step 2.4.2.1

Add $6$ to both sides of the equation.

$5 x = 6$

Step 2.4.2.2

Divide each term in $5 x = 6$ by $5$ and simplify.

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Step 2.4.2.2.1

Divide each term in $5 x = 6$ by $5$ .

$\frac{5 x}{5} = \frac{6}{5}$

Step 2.4.2.2.2

Simplify the left side.

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Step 2.4.2.2.2.1

Cancel the common factor of $5$ .

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Step 2.4.2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{6}{5}$

Step 2.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{6}{5}$

Step 2.5

The final solution is all the values that make $3 (4 x + 1) (5 x - 6) = 0$ true.

$x = - \frac{1}{4}, \frac{6}{5}$

Step 3