Algebra Examples

$f (x) = 2 x^{2} - x - 1$ , $f (x) = - 1$

Step 1

Substitute $- 1$ for $f (x)$ .

$- 1 = 2 x^{2} - x - 1$

Step 2

Solve for $x$ .

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Step 2.1

Rewrite the equation as $2 x^{2} - x - 1 = - 1$ .

$2 x^{2} - x - 1 = - 1$

Step 2.2

Add $1$ to both sides of the equation.

$2 x^{2} - x - 1 + 1 = 0$

Step 2.3

Combine the opposite terms in $2 x^{2} - x - 1 + 1$ .

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Step 2.3.1

Add $- 1$ and $1$ .

$2 x^{2} - x + 0 = 0$

Step 2.3.2

Add $2 x^{2} - x$ and $0$ .

$2 x^{2} - x = 0$

Step 2.4

Factor $x$ out of $2 x^{2} - x$ .

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Step 2.4.1

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

$x (2 x) - x = 0$

Step 2.4.2

Factor $x$ out of $- x$ .

$x (2 x) + x \cdot - 1 = 0$

Step 2.4.3

Factor $x$ out of $x (2 x) + x \cdot - 1$ .

$x (2 x - 1) = 0$

Step 2.5

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

$x = 0$

$2 x - 1 = 0$

Step 2.6

Set $x$ equal to $0$ .

$x = 0$

Step 2.7

Set $2 x - 1$ equal to $0$ and solve for $x$ .

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Step 2.7.1

Set $2 x - 1$ equal to $0$ .

$2 x - 1 = 0$

Step 2.7.2

Solve $2 x - 1 = 0$ for $x$ .

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Step 2.7.2.1

Add $1$ to both sides of the equation.

$2 x = 1$

Step 2.7.2.2

Divide each term in $2 x = 1$ by $2$ and simplify.

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Step 2.7.2.2.1

Divide each term in $2 x = 1$ by $2$ .

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

Step 2.7.2.2.2

Simplify the left side.

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Step 2.7.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.7.2.2.2.1.1

Cancel the common factor.

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

Step 2.7.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 2.8

The final solution is all the values that make $x (2 x - 1) = 0$ true.

$x = 0, \frac{1}{2}$