Algebra Examples

$f (x) = x^{2} - 5 x + 3$ , $f (x) = - 3$

Step 1

Eliminate the equal sides of each equation and combine.

$x^{2} - 5 x + 3 = - 3$

Step 2

Solve $x^{2} - 5 x + 3 = - 3$ for $x$ .

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Step 2.1

Add $3$ to both sides of the equation.

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

Step 2.2

Add $3$ and $3$ .

$x^{2} - 5 x + 6 = 0$

Step 2.3

Factor $x^{2} - 5 x + 6$ using the AC method.

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Step 2.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $6$ and whose sum is $- 5$ .

$- 3, - 2 + f (x) = - 3$

Step 2.3.2

Write the factored form using these integers.

$(x - 3) (x - 2) = 0$

Step 2.4

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

$x - 3 = 0$

$x - 2 = 0 + f (x) = - 3$

Step 2.5

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

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Step 2.5.1

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

$x - 3 = 0$

Step 2.5.2

Add $3$ to both sides of the equation.

$x = 3$

Step 2.6

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

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Step 2.6.1

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

$x - 2 = 0$

Step 2.6.2

Add $2$ to both sides of the equation.

$x = 2$

Step 2.7

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

$x = 3, 2$

Step 3

Evaluate $f$ when $x = 3$ .

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Step 3.1

Substitute $3$ for $x$ .

$f (3) = - 3$

Step 3.2

Divide each term in $f (3) = - 3$ by $3$ and simplify.

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Step 3.2.1

Divide each term in $f (3) = - 3$ by $3$ .

$\frac{f \cdot 3}{3} = \frac{- 3}{3}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $3$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{f \cdot 3}{3} = \frac{- 3}{3}$

Step 3.2.2.1.2

Divide $f$ by $1$ .

$f = \frac{- 3}{3}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $- 3$ by $3$ .

$f = - 1$

Step 4

Evaluate $f$ when $x = 2$ .

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Step 4.1

Substitute $2$ for $x$ .

$f (2) = - 3$

Step 4.2

Divide each term in $f (2) = - 3$ by $2$ and simplify.

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Step 4.2.1

Divide each term in $f (2) = - 3$ by $2$ .

$\frac{f \cdot 2}{2} = \frac{- 3}{2}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $2$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{f \cdot 2}{2} = \frac{- 3}{2}$

Step 4.2.2.1.2

Divide $f$ by $1$ .

$f = \frac{- 3}{2}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Move the negative in front of the fraction.

$f = - \frac{3}{2}$

Step 5

The solution of the system of equations is all the values that make the system true.

$f = - 1, x = 3 f = - \frac{3}{2}, x = 2$

Step 6

List all of the solutions.

$f = - 1, x = 3$

$f = - \frac{3}{2}, x = 2$