Calculus Examples

${(x^{2} - 2 x)}^{2} - 11 (x^{2} - 2 x) + 24 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Let $u = x^{2} - 2 x$ . Substitute $u$ for all occurrences of $x^{2} - 2 x$ .

$u^{2} - 11 u + 24 = 0$

Step 1.2

Factor $u^{2} - 11 u + 24$ 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 $24$ and whose sum is $- 11$ .

$- 8, - 3$

Step 1.2.2

Write the factored form using these integers.

$(u - 8) (u - 3) = 0$

Step 1.3

Replace all occurrences of $u$ with $x^{2} - 2 x$ .

$(x^{2} - 2 x - 8) (x^{2} - 2 x - 3) = 0$

Step 2

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

$x^{2} - 2 x - 8 = 0$

$x^{2} - 2 x - 3 = 0$

Step 3

Set $x^{2} - 2 x - 8$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $x^{2} - 2 x - 8$ equal to $0$ .

$x^{2} - 2 x - 8 = 0$

Step 3.2

Solve $x^{2} - 2 x - 8 = 0$ for $x$ .

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Step 3.2.1

Factor $x^{2} - 2 x - 8$ using the AC method.

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Step 3.2.1.1

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 $- 8$ and whose sum is $- 2$ .

$- 4, 2$

Step 3.2.1.2

Write the factored form using these integers.

$(x - 4) (x + 2) = 0$

Step 3.2.2

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

$x - 4 = 0$

$x + 2 = 0$

Step 3.2.3

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

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Step 3.2.3.1

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

$x - 4 = 0$

Step 3.2.3.2

Add $4$ to both sides of the equation.

$x = 4$

Step 3.2.4

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

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Step 3.2.4.1

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

$x + 2 = 0$

Step 3.2.4.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 3.2.5

The final solution is all the values that make $(x - 4) (x + 2) = 0$ true.

$x = 4, - 2$

Step 4

Set $x^{2} - 2 x - 3$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x^{2} - 2 x - 3$ equal to $0$ .

$x^{2} - 2 x - 3 = 0$

Step 4.2

Solve $x^{2} - 2 x - 3 = 0$ for $x$ .

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Step 4.2.1

Factor $x^{2} - 2 x - 3$ 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 $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Step 4.2.1.2

Write the factored form using these integers.

$(x - 3) (x + 1) = 0$

Step 4.2.2

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

$x - 3 = 0$

$x + 1 = 0$

Step 4.2.3

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

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Step 4.2.3.1

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

$x - 3 = 0$

Step 4.2.3.2

Add $3$ to both sides of the equation.

$x = 3$

Step 4.2.4

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 4.2.4.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 4.2.4.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 4.2.5

The final solution is all the values that make $(x - 3) (x + 1) = 0$ true.

$x = 3, - 1$

Step 5

The final solution is all the values that make $(x^{2} - 2 x - 8) (x^{2} - 2 x - 3) = 0$ true.

$x = 4, - 2, 3, - 1$