Precalculus Examples

$z (2 z + 7) = 4$

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Simplify the left side.

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Step 1.1.1

Simplify $z (2 z + 7)$ .

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Step 1.1.1.1

Simplify by multiplying through.

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Step 1.1.1.1.1

Apply the distributive property.

$z (2 z) + z \cdot 7 = 4$

Step 1.1.1.1.2

Reorder.

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Step 1.1.1.1.2.1

Rewrite using the commutative property of multiplication.

$2 z \cdot z + z \cdot 7 = 4$

Step 1.1.1.1.2.2

Move $7$ to the left of $z$ .

$2 z \cdot z + 7 \cdot z = 4$

Step 1.1.1.2

Multiply $z$ by $z$ by adding the exponents.

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Step 1.1.1.2.1

Move $z$ .

$2 (z \cdot z) + 7 \cdot z = 4$

Step 1.1.1.2.2

Multiply $z$ by $z$ .

$2 z^{2} + 7 \cdot z = 4$

$2 z^{2} + 7 z = 4$

Step 1.2

Subtract $4$ from both sides of the equation.

$2 z^{2} + 7 z - 4 = 0$

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 $a = 2$ , $b = 7$ , and $c = - 4$ into the quadratic formula and solve for $z$ .

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $7$ to the power of $2$ .

$z = \frac{- 7 \pm \sqrt{49 - 4 \cdot 2 \cdot - 4}}{2 \cdot 2}$

Step 4.1.2

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

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Step 4.1.2.1

Multiply $- 4$ by $2$ .

$z = \frac{- 7 \pm \sqrt{49 - 8 \cdot - 4}}{2 \cdot 2}$

Step 4.1.2.2

Multiply $- 8$ by $- 4$ .

$z = \frac{- 7 \pm \sqrt{49 + 32}}{2 \cdot 2}$

Step 4.1.3

Add $49$ and $32$ .

$z = \frac{- 7 \pm \sqrt{81}}{2 \cdot 2}$

Step 4.1.4

Rewrite $81$ as $9^{2}$ .

$z = \frac{- 7 \pm \sqrt{9^{2}}}{2 \cdot 2}$

Step 4.1.5

Pull terms out from under the radical, assuming positive real numbers.

$z = \frac{- 7 \pm 9}{2 \cdot 2}$

Step 4.2

Multiply $2$ by $2$ .

$z = \frac{- 7 \pm 9}{4}$

Step 5

The final answer is the combination of both solutions.

$z = \frac{1}{2}, - 4$