Algebra Examples

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

Step 1

Subtract $t$ from both sides of the equation.

$2 x^{2} - 4 x - t = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 2$ , $b = - 4$ , and $c = - t$ into the quadratic formula and solve for $x$ .

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Factor $- 4$ out of ${(- 4)}^{2} - 4 \cdot 2 \cdot - 1 t$ .

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Step 4.1.1.1

Factor $- 4$ out of ${(- 4)}^{2}$ .

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

Step 4.1.1.2

Factor $- 4$ out of $- 4 \cdot 2 \cdot - 1 t$ .

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

Step 4.1.1.3

Factor $- 4$ out of $- 4 \cdot - 4 - 4 (2 \cdot - 1 t)$ .

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

Step 4.1.2

Multiply $2$ by $- 1$ .

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

Step 4.1.3

Factor $2$ out of $- 4 - 2 t$ .

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Step 4.1.3.1

Factor $2$ out of $- 4$ .

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

Step 4.1.3.2

Factor $2$ out of $- 2 t$ .

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

Step 4.1.3.3

Factor $2$ out of $2 (- 2) + 2 (- t)$ .

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

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

Step 4.1.4

Multiply $- 4$ by $2$ .

$x = \frac{4 \pm \sqrt{- 8 (- 2 - t)}}{2 \cdot 2}$

Step 4.1.5

Rewrite $- 8 (- 2 - t)$ as $2^{2} \cdot (- 2 (- 2 - t))$ .

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Step 4.1.5.1

Factor $4$ out of $- 8$ .

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

Step 4.1.5.2

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

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

Step 4.1.5.3

Add parentheses.

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

Step 4.1.6

Pull terms out from under the radical.

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

Step 4.2

Multiply $2$ by $2$ .

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

Step 4.3

Simplify $\frac{4 \pm 2 \sqrt{- 2 (- 2 - t)}}{4}$ .

$x = \frac{2 \pm \sqrt{- 2 (- 2 - t)}}{2}$

Step 5

The final answer is the combination of both solutions.

$x = \frac{2 + \sqrt{- 2 (- 2 - t)}}{2}$

$x = \frac{2 - \sqrt{- 2 (- 2 - t)}}{2}$