Calculus Examples

$x^{2} - x - 7 = x + 8$

Step 1

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

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Step 1.1

Subtract $x$ from both sides of the equation.

$x^{2} - x - 7 - x = 8$

Step 1.2

Subtract $x$ from $- x$ .

$x^{2} - 2 x - 7 = 8$

Step 2

Subtract $8$ from both sides of the equation.

$x^{2} - 2 x - 7 - 8 = 0$

Step 3

Subtract $8$ from $- 7$ .

$x^{2} - 2 x - 15 = 0$

Step 4

Factor $x^{2} - 2 x - 15$ using the AC method.

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Step 4.1

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 $- 2$ .

$- 5, 3$

Step 4.2

Write the factored form using these integers.

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

Step 5

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

$x - 5 = 0$

$x + 3 = 0$

Step 6

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

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Step 6.1

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

$x - 5 = 0$

Step 6.2

Add $5$ to both sides of the equation.

$x = 5$

Step 7

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

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Step 7.1

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

$x + 3 = 0$

Step 7.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 8

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

$x = 5, - 3$