Calculus Examples

$e^{t^{2}} = e^{4 t - 3}$

Step 1

Create equivalent expressions in the equation that all have equal bases.

$e^{t^{2}} = e^{4 t - 3}$

Step 2

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$t^{2} = 4 t - 3$

Step 3

Solve for $t$ .

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Step 3.1

Subtract $4 t$ from both sides of the equation.

$t^{2} - 4 t = - 3$

Step 3.2

Add $3$ to both sides of the equation.

$t^{2} - 4 t + 3 = 0$

Step 3.3

Factor $t^{2} - 4 t + 3$ using the AC method.

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Step 3.3.1

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 $- 4$ .

$- 3, - 1$

Step 3.3.2

Write the factored form using these integers.

$(t - 3) (t - 1) = 0$

Step 3.4

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

$t - 3 = 0$

$t - 1 = 0$

Step 3.5

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

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Step 3.5.1

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

$t - 3 = 0$

Step 3.5.2

Add $3$ to both sides of the equation.

$t = 3$

Step 3.6

Set $t - 1$ equal to $0$ and solve for $t$ .

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Step 3.6.1

Set $t - 1$ equal to $0$ .

$t - 1 = 0$

Step 3.6.2

Add $1$ to both sides of the equation.

$t = 1$

Step 3.7

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

$t = 3, 1$