Algebra Examples

$0 = x^{2} - 4 x - 60$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $x^{2}$ from both sides of the equation.

$- x^{2} = - 4 x - 60$

Step 1.2

Add $4 x$ to both sides of the equation.

$- x^{2} + 4 x = - 60$

Step 1.3

Add $60$ to both sides of the equation.

$- x^{2} + 4 x + 60 = 0$

Step 2

Factor by grouping.

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Step 2.1

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 $a \cdot c = - 1 \cdot 60 = - 60$ and whose sum is $b = 4$ .

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Step 2.1.1

Factor $4$ out of $4 x$ .

$- x^{2} + 4 (x) + 60 = 0$

Step 2.1.2

Rewrite $4$ as $- 6$ plus $10$

$- x^{2} + (- 6 + 10) x + 60 = 0$

Step 2.1.3

Apply the distributive property.

$- x^{2} - 6 x + 10 x + 60 = 0$

Step 2.2

Factor out the greatest common factor from each group.

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Step 2.2.1

Group the first two terms and the last two terms.

$(- x^{2} - 6 x) + 10 x + 60 = 0$

Step 2.2.2

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

$x (- x - 6) - 10 (- x - 6) = 0$

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $- x - 6$ .

$(- x - 6) (x - 10) = 0$

Step 3

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

$- x - 6 = 0$

$x - 10 = 0$

Step 4

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

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Step 4.1

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

$- x - 6 = 0$

Step 4.2

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

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Step 4.2.1

Add $6$ to both sides of the equation.

$- x = 6$

Step 4.2.2

Divide each term in $- x = 6$ by $- 1$ and simplify.

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Step 4.2.2.1

Divide each term in $- x = 6$ by $- 1$ .

$\frac{- x}{- 1} = \frac{6}{- 1}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{6}{- 1}$

Step 4.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{6}{- 1}$

Step 4.2.2.3

Simplify the right side.

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Step 4.2.2.3.1

Divide $6$ by $- 1$ .

$x = - 6$

Step 5

Set $x - 10$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $x - 10$ equal to $0$ .

$x - 10 = 0$

Step 5.2

Add $10$ to both sides of the equation.

$x = 10$

Step 6

The final solution is all the values that make $(- x - 6) (x - 10) = 0$ true.

$x = - 6, 10$