Pre-Algebra Examples

$t^{2} + 2 t - 1 = 0$

Step 1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2

Substitute the values $a = 1$ , $b = 2$ , and $c = - 1$ into the quadratic formula and solve for $t$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (1 \cdot - 1)}}{2 \cdot 1}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $2$ to the power of $2$ .

$t = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 1}}{2 \cdot 1}$

Step 3.1.2

Multiply $- 4 \cdot 1 \cdot - 1$ .

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Step 3.1.2.1

Multiply $- 4$ by $1$ .

$t = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 1}}{2 \cdot 1}$

Step 3.1.2.2

Multiply $- 4$ by $- 1$ .

$t = \frac{- 2 \pm \sqrt{4 + 4}}{2 \cdot 1}$

Step 3.1.3

Add $4$ and $4$ .

$t = \frac{- 2 \pm \sqrt{8}}{2 \cdot 1}$

Step 3.1.4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 3.1.4.1

Factor $4$ out of $8$ .

$t = \frac{- 2 \pm \sqrt{4 (2)}}{2 \cdot 1}$

Step 3.1.4.2

Rewrite $4$ as $2^{2}$ .

$t = \frac{- 2 \pm \sqrt{2^{2} \cdot 2}}{2 \cdot 1}$

Step 3.1.5

Pull terms out from under the radical.

$t = \frac{- 2 \pm 2 \sqrt{2}}{2 \cdot 1}$

Step 3.2

Multiply $2$ by $1$ .

$t = \frac{- 2 \pm 2 \sqrt{2}}{2}$

Step 3.3

Simplify $\frac{- 2 \pm 2 \sqrt{2}}{2}$ .

$t = - 1 \pm \sqrt{2}$

Step 4

The final answer is the combination of both solutions.

$t = - 1 + \sqrt{2}, - 1 - \sqrt{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$t = - 1 + \sqrt{2}, - 1 - \sqrt{2}$

Decimal Form:

$t = 0.41421356 \dots, - 2.41421356 \dots$