Pre-Algebra Examples

$3 x (3 x - 16) = - 64$

Step 1

Divide each term in $3 x (3 x - 16) = - 64$ by $3$ and simplify.

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Step 1.1

Divide each term in $3 x (3 x - 16) = - 64$ by $3$ .

$\frac{3 x (3 x - 16)}{3} = \frac{- 64}{3}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Simplify terms.

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Step 1.2.1.1

Cancel the common factor of $3$ .

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Step 1.2.1.1.1

Cancel the common factor.

$\frac{3 x (3 x - 16)}{3} = \frac{- 64}{3}$

Step 1.2.1.1.2

Divide $x (3 x - 16)$ by $1$ .

$x (3 x - 16) = \frac{- 64}{3}$

Step 1.2.1.2

Apply the distributive property.

$x (3 x) + x \cdot - 16 = \frac{- 64}{3}$

Step 1.2.1.3

Reorder.

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Step 1.2.1.3.1

Rewrite using the commutative property of multiplication.

$3 x \cdot x + x \cdot - 16 = \frac{- 64}{3}$

Step 1.2.1.3.2

Move $- 16$ to the left of $x$ .

$3 x \cdot x - 16 \cdot x = \frac{- 64}{3}$

Step 1.2.2

Multiply $x$ by $x$ by adding the exponents.

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Step 1.2.2.1

Move $x$ .

$3 (x \cdot x) - 16 \cdot x = \frac{- 64}{3}$

Step 1.2.2.2

Multiply $x$ by $x$ .

$3 x^{2} - 16 \cdot x = \frac{- 64}{3}$

$3 x^{2} - 16 x = \frac{- 64}{3}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Move the negative in front of the fraction.

$3 x^{2} - 16 x = - \frac{64}{3}$

Step 2

Add $\frac{64}{3}$ to both sides of the equation.

$3 x^{2} - 16 x + \frac{64}{3} = 0$

Step 3

Multiply through by the least common denominator $3$ , then simplify.

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Step 3.1

Apply the distributive property.

$3 (3 x^{2}) + 3 (- 16 x) + 3 (\frac{64}{3}) = 0$

Step 3.2

Simplify.

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Step 3.2.1

Multiply $3$ by $3$ .

$9 x^{2} + 3 (- 16 x) + 3 (\frac{64}{3}) = 0$

Step 3.2.2

Multiply $- 16$ by $3$ .

$9 x^{2} - 48 x + 3 (\frac{64}{3}) = 0$

Step 3.2.3

Cancel the common factor of $3$ .

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Step 3.2.3.1

Cancel the common factor.

$9 x^{2} - 48 x + 3 (\frac{64}{3}) = 0$

Step 3.2.3.2

Rewrite the expression.

$9 x^{2} - 48 x + 64 = 0$

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = 9$ , $b = - 48$ , and $c = 64$ into the quadratic formula and solve for $x$ .

$\frac{48 \pm \sqrt{{(- 48)}^{2} - 4 \cdot (9 \cdot 64)}}{2 \cdot 9}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $- 48$ to the power of $2$ .

$x = \frac{48 \pm \sqrt{2304 - 4 \cdot 9 \cdot 64}}{2 \cdot 9}$

Step 6.1.2

Multiply $- 4 \cdot 9 \cdot 64$ .

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Step 6.1.2.1

Multiply $- 4$ by $9$ .

$x = \frac{48 \pm \sqrt{2304 - 36 \cdot 64}}{2 \cdot 9}$

Step 6.1.2.2

Multiply $- 36$ by $64$ .

$x = \frac{48 \pm \sqrt{2304 - 2304}}{2 \cdot 9}$

Step 6.1.3

Subtract $2304$ from $2304$ .

$x = \frac{48 \pm \sqrt{0}}{2 \cdot 9}$

Step 6.1.4

Rewrite $0$ as $0^{2}$ .

$x = \frac{48 \pm \sqrt{0^{2}}}{2 \cdot 9}$

Step 6.1.5

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

$x = \frac{48 \pm 0}{2 \cdot 9}$

Step 6.1.6

$48$ plus or minus $0$ is $48$ .

$x = \frac{48}{2 \cdot 9}$

Step 6.2

Multiply $2$ by $9$ .

$x = \frac{48}{18}$

Step 6.3

Cancel the common factor of $48$ and $18$ .

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Step 6.3.1

Factor $6$ out of $48$ .

$x = \frac{6 (8)}{18}$

Step 6.3.2

Cancel the common factors.

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Step 6.3.2.1

Factor $6$ out of $18$ .

$x = \frac{6 \cdot 8}{6 \cdot 3}$

Step 6.3.2.2

Cancel the common factor.

$x = \frac{6 \cdot 8}{6 \cdot 3}$

Step 6.3.2.3

Rewrite the expression.

$x = \frac{8}{3}$

Step 7

The final answer is the combination of both solutions.

$x = \frac{8}{3}$ Double roots