Precalculus Examples

$3 x (x - 1) - x (x - 8) = 3$

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 $3 x (x - 1) - x (x - 8)$ .

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Step 1.1.1.1

Simplify each term.

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Step 1.1.1.1.1

Apply the distributive property.

$3 x \cdot x + 3 x \cdot - 1 - x (x - 8) = 3$

Step 1.1.1.1.2

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

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Step 1.1.1.1.2.1

Move $x$ .

$3 (x \cdot x) + 3 x \cdot - 1 - x (x - 8) = 3$

Step 1.1.1.1.2.2

Multiply $x$ by $x$ .

$3 x^{2} + 3 x \cdot - 1 - x (x - 8) = 3$

Step 1.1.1.1.3

Multiply $- 1$ by $3$ .

$3 x^{2} - 3 x - x (x - 8) = 3$

Step 1.1.1.1.4

Apply the distributive property.

$3 x^{2} - 3 x - x \cdot x - x \cdot - 8 = 3$

Step 1.1.1.1.5

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

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Step 1.1.1.1.5.1

Move $x$ .

$3 x^{2} - 3 x - (x \cdot x) - x \cdot - 8 = 3$

Step 1.1.1.1.5.2

Multiply $x$ by $x$ .

$3 x^{2} - 3 x - x^{2} - x \cdot - 8 = 3$

Step 1.1.1.1.6

Multiply $- 8$ by $- 1$ .

$3 x^{2} - 3 x - x^{2} + 8 x = 3$

Step 1.1.1.2

Simplify by adding terms.

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Step 1.1.1.2.1

Subtract $x^{2}$ from $3 x^{2}$ .

$2 x^{2} - 3 x + 8 x = 3$

Step 1.1.1.2.2

Add $- 3 x$ and $8 x$ .

$2 x^{2} + 5 x = 3$

Step 1.2

Subtract $3$ from both sides of the equation.

$2 x^{2} + 5 x - 3 = 0$

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 $a = 2$ , $b = 5$ , and $c = - 3$ into the quadratic formula and solve for $x$ .

$\frac{- 5 \pm \sqrt{5^{2} - 4 \cdot (2 \cdot - 3)}}{2 \cdot 2}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $5$ to the power of $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 2 \cdot - 3}}{2 \cdot 2}$

Step 4.1.2

Multiply $- 4 \cdot 2 \cdot - 3$ .

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Step 4.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 8 \cdot - 3}}{2 \cdot 2}$

Step 4.1.2.2

Multiply $- 8$ by $- 3$ .

$x = \frac{- 5 \pm \sqrt{25 + 24}}{2 \cdot 2}$

Step 4.1.3

Add $25$ and $24$ .

$x = \frac{- 5 \pm \sqrt{49}}{2 \cdot 2}$

Step 4.1.4

Rewrite $49$ as $7^{2}$ .

$x = \frac{- 5 \pm \sqrt{7^{2}}}{2 \cdot 2}$

Step 4.1.5

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

$x = \frac{- 5 \pm 7}{2 \cdot 2}$

Step 4.2

Multiply $2$ by $2$ .

$x = \frac{- 5 \pm 7}{4}$

Step 5

The final answer is the combination of both solutions.

$x = \frac{1}{2}, - 3$