Algebra Examples

$y = - \frac{1}{2} x^{2}$ $y = x - 4$

Step 1

Eliminate the equal sides of each equation and combine.

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

Step 2

Solve $- \frac{1}{2} x^{2} = x - 4$ for $x$ .

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Step 2.1

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

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Step 2.1.1

Rewrite.

$0 + 0 - \frac{1}{2} x^{2} = x - 4$

Step 2.1.2

Simplify by adding zeros.

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

Step 2.1.3

Combine $x^{2}$ and $\frac{1}{2}$ .

$- \frac{x^{2}}{2} = x - 4$

Step 2.2

Subtract $x$ from both sides of the equation.

$- \frac{x^{2}}{2} - x = - 4$

Step 2.3

Multiply each term in $- \frac{x^{2}}{2} - x = - 4$ by $2$ to eliminate the fractions.

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Step 2.3.1

Multiply each term in $- \frac{x^{2}}{2} - x = - 4$ by $2$ .

$- \frac{x^{2}}{2} \cdot 2 - x \cdot 2 = - 4 \cdot 2$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Simplify each term.

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Step 2.3.2.1.1

Cancel the common factor of $2$ .

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Step 2.3.2.1.1.1

Move the leading negative in $- \frac{x^{2}}{2}$ into the numerator.

$\frac{- x^{2}}{2} \cdot 2 - x \cdot 2 = - 4 \cdot 2$

Step 2.3.2.1.1.2

Cancel the common factor.

$\frac{- x^{2}}{2} \cdot 2 - x \cdot 2 = - 4 \cdot 2$

Step 2.3.2.1.1.3

Rewrite the expression.

$- x^{2} - x \cdot 2 = - 4 \cdot 2$

Step 2.3.2.1.2

Multiply $2$ by $- 1$ .

$- x^{2} - 2 x = - 4 \cdot 2$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Multiply $- 4$ by $2$ .

$- x^{2} - 2 x = - 8$

Step 2.4

Add $8$ to both sides of the equation.

$- x^{2} - 2 x + 8 = 0$

Step 2.5

Factor the left side of the equation.

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Step 2.5.1

Factor $- 1$ out of $- x^{2} - 2 x + 8$ .

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Step 2.5.1.1

Factor $- 1$ out of $- x^{2}$ .

$- (x^{2}) - 2 x + 8 = 0$

Step 2.5.1.2

Factor $- 1$ out of $- 2 x$ .

$- (x^{2}) - (2 x) + 8 = 0$

Step 2.5.1.3

Rewrite $8$ as $- 1 (- 8)$ .

$- (x^{2}) - (2 x) - 1 \cdot - 8 = 0$

Step 2.5.1.4

Factor $- 1$ out of $- (x^{2}) - (2 x)$ .

$- (x^{2} + 2 x) - 1 \cdot - 8 = 0$

Step 2.5.1.5

Factor $- 1$ out of $- (x^{2} + 2 x) - 1 (- 8)$ .

$- (x^{2} + 2 x - 8) = 0$

Step 2.5.2

Factor.

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Step 2.5.2.1

Factor $x^{2} + 2 x - 8$ using the AC method.

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Step 2.5.2.1.1

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 $- 8$ and whose sum is $2$ .

$- 2, 4 + y = x - 4$

Step 2.5.2.1.2

Write the factored form using these integers.

$- ((x - 2) (x + 4)) = 0$

Step 2.5.2.2

Remove unnecessary parentheses.

$- (x - 2) (x + 4) = 0$

Step 2.6

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

$x - 2 = 0$

$x + 4 = 0 + y = x - 4$

Step 2.7

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

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Step 2.7.1

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

$x - 2 = 0$

Step 2.7.2

Add $2$ to both sides of the equation.

$x = 2$

Step 2.8

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

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Step 2.8.1

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

$x + 4 = 0$

Step 2.8.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 2.9

The final solution is all the values that make $- (x - 2) (x + 4) = 0$ true.

$x = 2, - 4$

Step 3

Evaluate $y$ when $x = 2$ .

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Step 3.1

Substitute $2$ for $x$ .

$y = (2) - 4$

Step 3.2

Substitute $2$ for $x$ in $y = (2) - 4$ and solve for $y$ .

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Step 3.2.1

Remove parentheses.

$y = 2 - 4$

Step 3.2.2

Remove parentheses.

$y = (2) - 4$

Step 3.2.3

Subtract $4$ from $2$ .

$y = - 2$

Step 4

Evaluate $y$ when $x = - 4$ .

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Step 4.1

Substitute $- 4$ for $x$ .

$y = (- 4) - 4$

Step 4.2

Substitute $- 4$ for $x$ in $y = (- 4) - 4$ and solve for $y$ .

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Step 4.2.1

Remove parentheses.

$y = - 4 - 4$

Step 4.2.2

Remove parentheses.

$y = (- 4) - 4$

Step 4.2.3

Subtract $4$ from $- 4$ .

$y = - 8$

Step 5

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(2, - 2)$

$(- 4, - 8)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(2, - 2), (- 4, - 8)$

Equation Form:

$x = 2, y = - 2$

$x = - 4, y = - 8$

Step 7