Finite Math Examples

${(x - 7)}^{2} - 5 (x - 7) + 4 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Let $u = x - 7$ . Substitute $u$ for all occurrences of $x - 7$ .

$u^{2} - 5 u + 4 = 0$

Step 1.2

Factor $u^{2} - 5 u + 4$ 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 $4$ and whose sum is $- 5$ .

$- 4, - 1$

Step 1.2.2

Write the factored form using these integers.

$(u - 4) (u - 1) = 0$

Step 1.3

Replace all occurrences of $u$ with $x - 7$ .

$(x - 7 - 4) (x - 7 - 1) = 0$

Step 2

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

$x - 7 - 4 = 0$

$x - 7 - 1 = 0$

Step 3

Set $x - 7 - 4$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $x - 7 - 4$ equal to $0$ .

$x - 7 - 4 = 0$

Step 3.2

Solve $x - 7 - 4 = 0$ for $x$ .

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Step 3.2.1

Subtract $4$ from $- 7$ .

$x - 11 = 0$

Step 3.2.2

Add $11$ to both sides of the equation.

$x = 11$

Step 4

Set $x - 7 - 1$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x - 7 - 1$ equal to $0$ .

$x - 7 - 1 = 0$

Step 4.2

Solve $x - 7 - 1 = 0$ for $x$ .

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Step 4.2.1

Subtract $1$ from $- 7$ .

$x - 8 = 0$

Step 4.2.2

Add $8$ to both sides of the equation.

$x = 8$

Step 5

The final solution is all the values that make $(x - 7 - 4) (x - 7 - 1) = 0$ true.

$x = 11, 8$