Algebra Examples

$- \frac{1}{2} x^{2} + 5 x = 8$

Step 1

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

$- \frac{x^{2}}{2} + 5 x = 8$

Step 2

Multiply each term in $- \frac{x^{2}}{2} + 5 x = 8$ by $2$ to eliminate the fractions.

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Step 2.1

Multiply each term in $- \frac{x^{2}}{2} + 5 x = 8$ by $2$ .

$- \frac{x^{2}}{2} \cdot 2 + 5 x \cdot 2 = 8 \cdot 2$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Cancel the common factor of $2$ .

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Step 2.2.1.1.1

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

$\frac{- x^{2}}{2} \cdot 2 + 5 x \cdot 2 = 8 \cdot 2$

Step 2.2.1.1.2

Cancel the common factor.

$\frac{- x^{2}}{2} \cdot 2 + 5 x \cdot 2 = 8 \cdot 2$

Step 2.2.1.1.3

Rewrite the expression.

$- x^{2} + 5 x \cdot 2 = 8 \cdot 2$

Step 2.2.1.2

Multiply $2$ by $5$ .

$- x^{2} + 10 x = 8 \cdot 2$

Step 2.3

Simplify the right side.

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Step 2.3.1

Multiply $8$ by $2$ .

$- x^{2} + 10 x = 16$

Step 3

Subtract $16$ from both sides of the equation.

$- x^{2} + 10 x - 16 = 0$

Step 4

Factor the left side of the equation.

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Step 4.1

Factor $- 1$ out of $- x^{2} + 10 x - 16$ .

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Step 4.1.1

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

$- (x^{2}) + 10 x - 16 = 0$

Step 4.1.2

Factor $- 1$ out of $10 x$ .

$- (x^{2}) - (- 10 x) - 16 = 0$

Step 4.1.3

Rewrite $- 16$ as $- 1 (16)$ .

$- (x^{2}) - (- 10 x) - 1 \cdot 16 = 0$

Step 4.1.4

Factor $- 1$ out of $- (x^{2}) - (- 10 x)$ .

$- (x^{2} - 10 x) - 1 \cdot 16 = 0$

Step 4.1.5

Factor $- 1$ out of $- (x^{2} - 10 x) - 1 (16)$ .

$- (x^{2} - 10 x + 16) = 0$

Step 4.2

Factor.

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Step 4.2.1

Factor $x^{2} - 10 x + 16$ using the AC method.

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Step 4.2.1.1

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 $16$ and whose sum is $- 10$ .

$- 8, - 2$

Step 4.2.1.2

Write the factored form using these integers.

$- ((x - 8) (x - 2)) = 0$

Step 4.2.2

Remove unnecessary parentheses.

$- (x - 8) (x - 2) = 0$

Step 5

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

$x - 8 = 0$

$x - 2 = 0$

Step 6

Set $x - 8$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $x - 8$ equal to $0$ .

$x - 8 = 0$

Step 6.2

Add $8$ to both sides of the equation.

$x = 8$

Step 7

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

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Step 7.1

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

$x - 2 = 0$

Step 7.2

Add $2$ to both sides of the equation.

$x = 2$

Step 8

The final solution is all the values that make $- (x - 8) (x - 2) = 0$ true.

$x = 8, 2$