Basic Math Examples

$z^{2} + (k - 4) z + 0.8 = 0$

Step 1

Apply the distributive property.

$z^{2} + k z - 4 z + 0.8 = 0$

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 $a = 1$ , $b = k - 4$ , and $c = 0.8$ into the quadratic formula and solve for $z$ .

$\frac{- (k - 4) \pm \sqrt{{(k - 4)}^{2} - 4 \cdot (1 \cdot 0.8)}}{2 \cdot 1}$

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.

$z = \frac{- k + 4 \pm \sqrt{{(k - 4)}^{2} - 4 \cdot 1 \cdot 0.8}}{2 \cdot 1}$

Step 4.1.2

Multiply $- 1$ by $- 4$ .

$z = \frac{- k + 4 \pm \sqrt{{(k - 4)}^{2} - 4 \cdot 1 \cdot 0.8}}{2 \cdot 1}$

Step 4.1.3

Rewrite $4 \cdot 1 \cdot 0.8$ as ${(2 \cdot 1 \cdot 0.89442719)}^{2}$ .

$z = \frac{- k + 4 \pm \sqrt{{(k - 4)}^{2} - {(2 \cdot 1 \cdot 0.89442719)}^{2}}}{2 \cdot 1}$

Step 4.1.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = k - 4$ and $b = 2 \cdot 1 \cdot 0.89442719$ .

$z = \frac{- k + 4 \pm \sqrt{(k - 4 + 2 \cdot 1 \cdot 0.89442719) (k - 4 - (2 \cdot 1 \cdot 0.89442719))}}{2 \cdot 1}$

Step 4.1.5

Simplify.

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Step 4.1.5.1

Multiply $2 \cdot 1 \cdot 0.89442719$ .

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Step 4.1.5.1.1

Multiply $2$ by $1$ .

$z = \frac{- k + 4 \pm \sqrt{(k - 4 + 2 \cdot 0.89442719) (k - 4 - (2 \cdot 1 \cdot 0.89442719))}}{2 \cdot 1}$

Step 4.1.5.1.2

Multiply $2$ by $0.89442719$ .

$z = \frac{- k + 4 \pm \sqrt{(k - 4 + 1.78885438) (k - 4 - (2 \cdot 1 \cdot 0.89442719))}}{2 \cdot 1}$

Step 4.1.5.2

Add $- 4$ and $1.78885438$ .

$z = \frac{- k + 4 \pm \sqrt{(k - 2.21114561) (k - 4 - (2 \cdot 1 \cdot 0.89442719))}}{2 \cdot 1}$

Step 4.1.5.3

Multiply $- (2 \cdot 1 \cdot 0.89442719)$ .

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Step 4.1.5.3.1

Multiply $2$ by $1$ .

$z = \frac{- k + 4 \pm \sqrt{(k - 2.21114561) (k - 4 - (2 \cdot 0.89442719))}}{2 \cdot 1}$

Step 4.1.5.3.2

Multiply $2$ by $0.89442719$ .

$z = \frac{- k + 4 \pm \sqrt{(k - 2.21114561) (k - 4 - 1 \cdot 1.78885438)}}{2 \cdot 1}$

Step 4.1.5.3.3

Multiply $- 1$ by $1.78885438$ .

$z = \frac{- k + 4 \pm \sqrt{(k - 2.21114561) (k - 4 - 1.78885438)}}{2 \cdot 1}$

Step 4.1.5.4

Subtract $1.78885438$ from $- 4$ .

$z = \frac{- k + 4 \pm \sqrt{(k - 2.21114561) (k - 5.78885438)}}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

$z = \frac{- k + 4 \pm \sqrt{(k - 2.21114561) (k - 5.78885438)}}{2}$

Step 5

The final answer is the combination of both solutions.

$z = - \frac{k - 4 - \sqrt{(k - 2.21114561) (k - 5.78885438)}}{2}$

$z = - \frac{k - 4 + \sqrt{(k - 2.21114561) (k - 5.78885438)}}{2}$