Basic Math Examples

$s = 7 t^{2} + t + 5$

Step 1

Rewrite the equation as $7 t^{2} + t + 5 = s$ .

$7 t^{2} + t + 5 = s$

Step 2

Subtract $s$ from both sides of the equation.

$7 t^{2} + t + 5 - s = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 7$ , $b = 1$ , and $c = 5 - s$ into the quadratic formula and solve for $t$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (7 \cdot (5 - s))}}{2 \cdot 7}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

One to any power is one.

$t = \frac{- 1 \pm \sqrt{1 - 4 \cdot 7 \cdot (5 - s)}}{2 \cdot 7}$

Step 5.1.2

Multiply $- 4$ by $7$ .

$t = \frac{- 1 \pm \sqrt{1 - 28 \cdot (5 - s)}}{2 \cdot 7}$

Step 5.1.3

Apply the distributive property.

$t = \frac{- 1 \pm \sqrt{1 - 28 \cdot 5 - 28 (- s)}}{2 \cdot 7}$

Step 5.1.4

Multiply $- 28$ by $5$ .

$t = \frac{- 1 \pm \sqrt{1 - 140 - 28 (- s)}}{2 \cdot 7}$

Step 5.1.5

Multiply $- 1$ by $- 28$ .

$t = \frac{- 1 \pm \sqrt{1 - 140 + 28 s}}{2 \cdot 7}$

Step 5.1.6

Subtract $140$ from $1$ .

$t = \frac{- 1 \pm \sqrt{- 139 + 28 s}}{2 \cdot 7}$

Step 5.2

Multiply $2$ by $7$ .

$t = \frac{- 1 \pm \sqrt{- 139 + 28 s}}{14}$

Step 6

The final answer is the combination of both solutions.

$t = - \frac{1 - \sqrt{- 139 + 28 s}}{14}$

$t = - \frac{1 + \sqrt{28 s - 139}}{14}$