Basic Math Examples

$b + b^{2} = 42$

Step 1

Subtract $42$ from both sides of the equation.

$b + b^{2} - 42 = 0$

Step 2

Factor the left side of the equation.

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Step 2.1

Let $u = b$ . Substitute $u$ for all occurrences of $b$ .

$u + u^{2} - 42 = 0$

Step 2.2

Factor $u + u^{2} - 42$ using the AC method.

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Step 2.2.1

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 $- 42$ and whose sum is $1$ .

$- 6, 7$

Step 2.2.2

Write the factored form using these integers.

$(u - 6) (u + 7) = 0$

Step 2.3

Replace all occurrences of $u$ with $b$ .

$(b - 6) (b + 7) = 0$

Step 3

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

$b - 6 = 0$

$b + 7 = 0$

Step 4

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

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Step 4.1

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

$b - 6 = 0$

Step 4.2

Add $6$ to both sides of the equation.

$b = 6$

Step 5

Set $b + 7$ equal to $0$ and solve for $b$ .

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Step 5.1

Set $b + 7$ equal to $0$ .

$b + 7 = 0$

Step 5.2

Subtract $7$ from both sides of the equation.

$b = - 7$

Step 6

The final solution is all the values that make $(b - 6) (b + 7) = 0$ true.

$b = 6, - 7$