Algebra Examples

$2 x^{2} - x + 55 = {(x + 5)}^{2}$

Step 1

Simplify ${(x + 5)}^{2}$ .

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Step 1.1

Rewrite.

$2 x^{2} - x + 55 = 0 + 0 + {(x + 5)}^{2}$

Step 1.2

Rewrite ${(x + 5)}^{2}$ as $(x + 5) (x + 5)$ .

$2 x^{2} - x + 55 = (x + 5) (x + 5)$

Step 1.3

Expand $(x + 5) (x + 5)$ using the FOIL Method.

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Step 1.3.1

Apply the distributive property.

$2 x^{2} - x + 55 = x (x + 5) + 5 (x + 5)$

Step 1.3.2

Apply the distributive property.

$2 x^{2} - x + 55 = x \cdot x + x \cdot 5 + 5 (x + 5)$

Step 1.3.3

Apply the distributive property.

$2 x^{2} - x + 55 = x \cdot x + x \cdot 5 + 5 x + 5 \cdot 5$

Step 1.4

Simplify and combine like terms.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Multiply $x$ by $x$ .

$2 x^{2} - x + 55 = x^{2} + x \cdot 5 + 5 x + 5 \cdot 5$

Step 1.4.1.2

Move $5$ to the left of $x$ .

$2 x^{2} - x + 55 = x^{2} + 5 \cdot x + 5 x + 5 \cdot 5$

Step 1.4.1.3

Multiply $5$ by $5$ .

$2 x^{2} - x + 55 = x^{2} + 5 x + 5 x + 25$

Step 1.4.2

Add $5 x$ and $5 x$ .

$2 x^{2} - x + 55 = x^{2} + 10 x + 25$

Step 2

Move all terms containing $x$ to the left side of the equation.

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Step 2.1

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

$2 x^{2} - x + 55 - x^{2} = 10 x + 25$

Step 2.2

Subtract $10 x$ from both sides of the equation.

$2 x^{2} - x + 55 - x^{2} - 10 x = 25$

Step 2.3

Subtract $x^{2}$ from $2 x^{2}$ .

$x^{2} - x + 55 - 10 x = 25$

Step 2.4

Subtract $10 x$ from $- x$ .

$x^{2} - 11 x + 55 = 25$

Step 3

Subtract $25$ from both sides of the equation.

$x^{2} - 11 x + 55 - 25 = 0$

Step 4

Subtract $25$ from $55$ .

$x^{2} - 11 x + 30 = 0$

Step 5

Factor $x^{2} - 11 x + 30$ using the AC method.

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Step 5.1

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 $30$ and whose sum is $- 11$ .

$- 6, - 5$

Step 5.2

Write the factored form using these integers.

$(x - 6) (x - 5) = 0$

Step 6

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

$x - 6 = 0$

$x - 5 = 0$

Step 7

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

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Step 7.1

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

$x - 6 = 0$

Step 7.2

Add $6$ to both sides of the equation.

$x = 6$

Step 8

Set $x - 5$ equal to $0$ and solve for $x$ .

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Step 8.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 8.2

Add $5$ to both sides of the equation.

$x = 5$

Step 9

The final solution is all the values that make $(x - 6) (x - 5) = 0$ true.

$x = 6, 5$