Precalculus Examples

$| 2 x - 5 | = 3 x^{2}$

Step 1

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$2 x - 5 = \pm 3 x^{2}$

Step 2

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.1

First, use the positive value of the $\pm$ to find the first solution.

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

Step 2.2

Subtract $3 x^{2}$ from both sides of the equation.

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

Step 2.3

Use the quadratic formula to find the solutions.

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

Step 2.4

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

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

Step 2.5

Simplify.

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

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

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

Step 2.5.1.2

Multiply $- 4 \cdot - 3 \cdot - 5$ .

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Step 2.5.1.2.1

Multiply $- 4$ by $- 3$ .

$x = \frac{- 2 \pm \sqrt{4 + 12 \cdot - 5}}{2 \cdot - 3}$

Step 2.5.1.2.2

Multiply $12$ by $- 5$ .

$x = \frac{- 2 \pm \sqrt{4 - 60}}{2 \cdot - 3}$

Step 2.5.1.3

Subtract $60$ from $4$ .

$x = \frac{- 2 \pm \sqrt{- 56}}{2 \cdot - 3}$

Step 2.5.1.4

Rewrite $- 56$ as $- 1 (56)$ .

$x = \frac{- 2 \pm \sqrt{- 1 \cdot 56}}{2 \cdot - 3}$

Step 2.5.1.5

Rewrite $\sqrt{- 1 (56)}$ as $\sqrt{- 1} \cdot \sqrt{56}$ .

$x = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{56}}{2 \cdot - 3}$

Step 2.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 2 \pm i \cdot \sqrt{56}}{2 \cdot - 3}$

Step 2.5.1.7

Rewrite $56$ as $2^{2} \cdot 14$ .

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Step 2.5.1.7.1

Factor $4$ out of $56$ .

$x = \frac{- 2 \pm i \cdot \sqrt{4 (14)}}{2 \cdot - 3}$

Step 2.5.1.7.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 2 \pm i \cdot \sqrt{2^{2} \cdot 14}}{2 \cdot - 3}$

Step 2.5.1.8

Pull terms out from under the radical.

$x = \frac{- 2 \pm i \cdot (2 \sqrt{14})}{2 \cdot - 3}$

Step 2.5.1.9

Move $2$ to the left of $i$ .

$x = \frac{- 2 \pm 2 i \sqrt{14}}{2 \cdot - 3}$

Step 2.5.2

Multiply $2$ by $- 3$ .

$x = \frac{- 2 \pm 2 i \sqrt{14}}{- 6}$

Step 2.5.3

Simplify $\frac{- 2 \pm 2 i \sqrt{14}}{- 6}$ .

$x = \frac{1 \pm i \sqrt{14}}{3}$

Step 2.6

The final answer is the combination of both solutions.

$x = \frac{1 + i \sqrt{14}}{3}, \frac{1 - i \sqrt{14}}{3}$

Step 2.7

Next, use the negative value of the $\pm$ to find the second solution.

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

Step 2.8

Add $3 x^{2}$ to both sides of the equation.

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

Step 2.9

Factor by grouping.

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Step 2.9.1

Reorder terms.

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

Step 2.9.2

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 5 = - 15$ and whose sum is $b = 2$ .

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Step 2.9.2.1

Factor $2$ out of $2 x$ .

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

Step 2.9.2.2

Rewrite $2$ as $- 3$ plus $5$

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

Step 2.9.2.3

Apply the distributive property.

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

Step 2.9.3

Factor out the greatest common factor from each group.

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Step 2.9.3.1

Group the first two terms and the last two terms.

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

Step 2.9.3.2

Factor out the greatest common factor (GCF) from each group.

$3 x (x - 1) + 5 (x - 1) = 0$

Step 2.9.4

Factor the polynomial by factoring out the greatest common factor, $x - 1$ .

$(x - 1) (3 x + 5) = 0$

Step 2.10

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 1 = 0$

$3 x + 5 = 0$

Step 2.11

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 2.11.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 2.11.2

Add $1$ to both sides of the equation.

$x = 1$

Step 2.12

Set $3 x + 5$ equal to $0$ and solve for $x$ .

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Step 2.12.1

Set $3 x + 5$ equal to $0$ .

$3 x + 5 = 0$

Step 2.12.2

Solve $3 x + 5 = 0$ for $x$ .

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Step 2.12.2.1

Subtract $5$ from both sides of the equation.

$3 x = - 5$

Step 2.12.2.2

Divide each term in $3 x = - 5$ by $3$ and simplify.

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Step 2.12.2.2.1

Divide each term in $3 x = - 5$ by $3$ .

$\frac{3 x}{3} = \frac{- 5}{3}$

Step 2.12.2.2.2

Simplify the left side.

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Step 2.12.2.2.2.1

Cancel the common factor of $3$ .

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Step 2.12.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 5}{3}$

Step 2.12.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 5}{3}$

Step 2.12.2.2.3

Simplify the right side.

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Step 2.12.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{5}{3}$

Step 2.13

The final solution is all the values that make $(x - 1) (3 x + 5) = 0$ true.

$x = 1, - \frac{5}{3}$

Step 2.14

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{1 + i \sqrt{14}}{3}, \frac{1 - i \sqrt{14}}{3}, 1, - \frac{5}{3}$