Pre-Algebra Examples

${(2 x - 5)}^{2} - 8 (2 x - 5) + 15 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Let $u = 2 x - 5$ . Substitute $u$ for all occurrences of $2 x - 5$ .

$u^{2} - 8 u + 15 = 0$

Step 1.2

Factor $u^{2} - 8 u + 15$ using the AC method.

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Step 1.2.1

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 $15$ and whose sum is $- 8$ .

$- 5, - 3$

Step 1.2.2

Write the factored form using these integers.

$(u - 5) (u - 3) = 0$

Step 1.3

Replace all occurrences of $u$ with $2 x - 5$ .

$(2 x - 5 - 5) (2 x - 5 - 3) = 0$

Step 2

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

$2 x - 5 - 5 = 0$

$2 x - 5 - 3 = 0$

Step 3

Set $2 x - 5 - 5$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $2 x - 5 - 5$ equal to $0$ .

$2 x - 5 - 5 = 0$

Step 3.2

Solve $2 x - 5 - 5 = 0$ for $x$ .

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Step 3.2.1

Subtract $5$ from $- 5$ .

$2 x - 10 = 0$

Step 3.2.2

Add $10$ to both sides of the equation.

$2 x = 10$

Step 3.2.3

Divide each term in $2 x = 10$ by $2$ and simplify.

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Step 3.2.3.1

Divide each term in $2 x = 10$ by $2$ .

$\frac{2 x}{2} = \frac{10}{2}$

Step 3.2.3.2

Simplify the left side.

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Step 3.2.3.2.1

Cancel the common factor of $2$ .

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Step 3.2.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{10}{2}$

Step 3.2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{10}{2}$

Step 3.2.3.3

Simplify the right side.

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Step 3.2.3.3.1

Divide $10$ by $2$ .

$x = 5$

Step 4

Set $2 x - 5 - 3$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $2 x - 5 - 3$ equal to $0$ .

$2 x - 5 - 3 = 0$

Step 4.2

Solve $2 x - 5 - 3 = 0$ for $x$ .

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Step 4.2.1

Subtract $3$ from $- 5$ .

$2 x - 8 = 0$

Step 4.2.2

Add $8$ to both sides of the equation.

$2 x = 8$

Step 4.2.3

Divide each term in $2 x = 8$ by $2$ and simplify.

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Step 4.2.3.1

Divide each term in $2 x = 8$ by $2$ .

$\frac{2 x}{2} = \frac{8}{2}$

Step 4.2.3.2

Simplify the left side.

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Step 4.2.3.2.1

Cancel the common factor of $2$ .

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Step 4.2.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{8}{2}$

Step 4.2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{8}{2}$

Step 4.2.3.3

Simplify the right side.

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Step 4.2.3.3.1

Divide $8$ by $2$ .

$x = 4$

Step 5

The final solution is all the values that make $(2 x - 5 - 5) (2 x - 5 - 3) = 0$ true.

$x = 5, 4$