Finite Math Examples

$w = 7 t^{2} - 266 t + 4200$

Step 1

Rewrite the equation as $7 t^{2} - 266 t + 4200 = w$ .

$7 t^{2} - 266 t + 4200 = w$

Step 2

Subtract $w$ from both sides of the equation.

$7 t^{2} - 266 t + 4200 - w = 0$

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 $a = 7$ , $b = - 266$ , and $c = 4200 - w$ into the quadratic formula and solve for $t$ .

$\frac{266 \pm \sqrt{{(- 266)}^{2} - 4 \cdot (7 \cdot (4200 - w))}}{2 \cdot 7}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 266$ to the power of $2$ .

$t = \frac{266 \pm \sqrt{70756 - 4 \cdot 7 \cdot (4200 - w)}}{2 \cdot 7}$

Step 5.1.2

Multiply $- 4$ by $7$ .

$t = \frac{266 \pm \sqrt{70756 - 28 \cdot (4200 - w)}}{2 \cdot 7}$

Step 5.1.3

Apply the distributive property.

$t = \frac{266 \pm \sqrt{70756 - 28 \cdot 4200 - 28 (- w)}}{2 \cdot 7}$

Step 5.1.4

Multiply $- 28$ by $4200$ .

$t = \frac{266 \pm \sqrt{70756 - 117600 - 28 (- w)}}{2 \cdot 7}$

Step 5.1.5

Multiply $- 1$ by $- 28$ .

$t = \frac{266 \pm \sqrt{70756 - 117600 + 28 w}}{2 \cdot 7}$

Step 5.1.6

Subtract $117600$ from $70756$ .

$t = \frac{266 \pm \sqrt{- 46844 + 28 w}}{2 \cdot 7}$

Step 5.1.7

Factor $28$ out of $- 46844 + 28 w$ .

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Step 5.1.7.1

Factor $28$ out of $- 46844$ .

$t = \frac{266 \pm \sqrt{28 \cdot - 1673 + 28 w}}{2 \cdot 7}$

Step 5.1.7.2

Factor $28$ out of $28 \cdot - 1673 + 28 w$ .

$t = \frac{266 \pm \sqrt{28 (- 1673 + w)}}{2 \cdot 7}$

Step 5.1.8

Rewrite $28 (- 1673 + w)$ as $2^{2} \cdot (7 (- 1673 + w))$ .

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Step 5.1.8.1

Factor $4$ out of $28$ .

$t = \frac{266 \pm \sqrt{4 (7) (- 1673 + w)}}{2 \cdot 7}$

Step 5.1.8.2

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

$t = \frac{266 \pm \sqrt{2^{2} \cdot (7 (- 1673 + w))}}{2 \cdot 7}$

Step 5.1.8.3

Add parentheses.

$t = \frac{266 \pm \sqrt{2^{2} \cdot (7 (- 1673 + w))}}{2 \cdot 7}$

Step 5.1.9

Pull terms out from under the radical.

$t = \frac{266 \pm 2 \sqrt{7 (- 1673 + w)}}{2 \cdot 7}$

Step 5.2

Multiply $2$ by $7$ .

$t = \frac{266 \pm 2 \sqrt{7 (- 1673 + w)}}{14}$

Step 5.3

Simplify $\frac{266 \pm 2 \sqrt{7 (- 1673 + w)}}{14}$ .

$t = \frac{133 \pm \sqrt{7 (- 1673 + w)}}{7}$

Step 6

The final answer is the combination of both solutions.

$t = \frac{133 + \sqrt{7 (- 1673 + w)}}{7}$

$t = \frac{133 - \sqrt{7 (- 1673 + w)}}{7}$