Calculus Examples

$2 x^{2} - (k - 3) x + 7 k = 0$

Step 1

Simplify each term.

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Step 1.1

Apply the distributive property.

$2 x^{2} + (- k - - 3) x + 7 k = 0$

Step 1.2

Multiply $- 1$ by $- 3$ .

$2 x^{2} + (- k + 3) x + 7 k = 0$

Step 1.3

Apply the distributive property.

$2 x^{2} - k x + 3 x + 7 k = 0$

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 $a = 2$ , $b = - k + 3$ , and $c = 7 k$ into the quadratic formula and solve for $x$ .

$\frac{- (- k + 3) \pm \sqrt{{(- k + 3)}^{2} - 4 \cdot (2 \cdot (7 k))}}{2 \cdot 2}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Apply the distributive property.

$x = \frac{k - 1 \cdot 3 \pm \sqrt{{(- k + 3)}^{2} - 4 \cdot 2 \cdot (7 k)}}{2 \cdot 2}$

Step 4.1.2

Multiply $- - k$ .

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Step 4.1.2.1

Multiply $- 1$ by $- 1$ .

$x = \frac{1 k - 1 \cdot 3 \pm \sqrt{{(- k + 3)}^{2} - 4 \cdot 2 \cdot (7 k)}}{2 \cdot 2}$

Step 4.1.2.2

Multiply $k$ by $1$ .

$x = \frac{k - 1 \cdot 3 \pm \sqrt{{(- k + 3)}^{2} - 4 \cdot 2 \cdot (7 k)}}{2 \cdot 2}$

Step 4.1.3

Multiply $- 1$ by $3$ .

$x = \frac{k - 3 \pm \sqrt{{(- k + 3)}^{2} - 4 \cdot 2 \cdot (7 k)}}{2 \cdot 2}$

Step 4.1.4

Rewrite ${(- k + 3)}^{2}$ as $(- k + 3) (- k + 3)$ .

$x = \frac{k - 3 \pm \sqrt{(- k + 3) (- k + 3) - 4 \cdot 2 \cdot (7 k)}}{2 \cdot 2}$

Step 4.1.5

Expand $(- k + 3) (- k + 3)$ using the FOIL Method.

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Step 4.1.5.1

Apply the distributive property.

$x = \frac{k - 3 \pm \sqrt{- k (- k + 3) + 3 (- k + 3) - 4 \cdot 2 \cdot (7 k)}}{2 \cdot 2}$

Step 4.1.5.2

Apply the distributive property.

$x = \frac{k - 3 \pm \sqrt{- k (- k) - k \cdot 3 + 3 (- k + 3) - 4 \cdot 2 \cdot (7 k)}}{2 \cdot 2}$

Step 4.1.5.3

Apply the distributive property.

$x = \frac{k - 3 \pm \sqrt{- k (- k) - k \cdot 3 + 3 (- k) + 3 \cdot 3 - 4 \cdot 2 \cdot (7 k)}}{2 \cdot 2}$

Step 4.1.6

Simplify and combine like terms.

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Step 4.1.6.1

Simplify each term.

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Step 4.1.6.1.1

Rewrite using the commutative property of multiplication.

$x = \frac{k - 3 \pm \sqrt{- 1 \cdot (- 1 k \cdot k) - k \cdot 3 + 3 (- k) + 3 \cdot 3 - 4 \cdot 2 \cdot (7 k)}}{2 \cdot 2}$

Step 4.1.6.1.2

Multiply $k$ by $k$ by adding the exponents.

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Step 4.1.6.1.2.1

Move $k$ .

$x = \frac{k - 3 \pm \sqrt{- 1 \cdot (- 1 (k \cdot k)) - k \cdot 3 + 3 (- k) + 3 \cdot 3 - 4 \cdot 2 \cdot (7 k)}}{2 \cdot 2}$

Step 4.1.6.1.2.2

Multiply $k$ by $k$ .

$x = \frac{k - 3 \pm \sqrt{- 1 \cdot (- 1 k^{2}) - k \cdot 3 + 3 (- k) + 3 \cdot 3 - 4 \cdot 2 \cdot (7 k)}}{2 \cdot 2}$

Step 4.1.6.1.3

Multiply $- 1$ by $- 1$ .

$x = \frac{k - 3 \pm \sqrt{1 k^{2} - k \cdot 3 + 3 (- k) + 3 \cdot 3 - 4 \cdot 2 \cdot (7 k)}}{2 \cdot 2}$

Step 4.1.6.1.4

Multiply $k^{2}$ by $1$ .

$x = \frac{k - 3 \pm \sqrt{k^{2} - k \cdot 3 + 3 (- k) + 3 \cdot 3 - 4 \cdot 2 \cdot (7 k)}}{2 \cdot 2}$

Step 4.1.6.1.5

Multiply $3$ by $- 1$ .

$x = \frac{k - 3 \pm \sqrt{k^{2} - 3 k + 3 (- k) + 3 \cdot 3 - 4 \cdot 2 \cdot (7 k)}}{2 \cdot 2}$

Step 4.1.6.1.6

Multiply $- 1$ by $3$ .

$x = \frac{k - 3 \pm \sqrt{k^{2} - 3 k - 3 k + 3 \cdot 3 - 4 \cdot 2 \cdot (7 k)}}{2 \cdot 2}$

Step 4.1.6.1.7

Multiply $3$ by $3$ .

$x = \frac{k - 3 \pm \sqrt{k^{2} - 3 k - 3 k + 9 - 4 \cdot 2 \cdot (7 k)}}{2 \cdot 2}$

Step 4.1.6.2

Subtract $3 k$ from $- 3 k$ .

$x = \frac{k - 3 \pm \sqrt{k^{2} - 6 k + 9 - 4 \cdot 2 \cdot (7 k)}}{2 \cdot 2}$

Step 4.1.7

Multiply $- 4 \cdot 2 \cdot 7$ .

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Step 4.1.7.1

Multiply $- 4$ by $2$ .

$x = \frac{k - 3 \pm \sqrt{k^{2} - 6 k + 9 - 8 \cdot (7 k)}}{2 \cdot 2}$

Step 4.1.7.2

Multiply $- 8$ by $7$ .

$x = \frac{k - 3 \pm \sqrt{k^{2} - 6 k + 9 - 56 k}}{2 \cdot 2}$

Step 4.1.8

Subtract $56 k$ from $- 6 k$ .

$x = \frac{k - 3 \pm \sqrt{k^{2} - 62 k + 9}}{2 \cdot 2}$

Step 4.2

Multiply $2$ by $2$ .

$x = \frac{k - 3 \pm \sqrt{k^{2} - 62 k + 9}}{4}$

Step 5

The final answer is the combination of both solutions.

$x = \frac{k - 3 + \sqrt{k^{2} - 62 k + 9}}{4}$

$x = \frac{k - 3 - \sqrt{k^{2} - 62 k + 9}}{4}$