Precalculus Examples Rational Expressions and Equations

Solve Over the Interval

$4 x^{2} + x = 3$ , $(0, 9)$

Step 1

Subtract $3$ from both sides of the equation.

$4 x^{2} + x - 3 = 0$

Step 2

Factor by grouping.

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Step 2.1

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 $a \cdot c = 4 \cdot - 3 = - 12$ and whose sum is $b = 1$ .

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Step 2.1.1

Multiply by $1$ .

$4 x^{2} + 1 x - 3 = 0$

Step 2.1.2

Rewrite $1$ as $- 3$ plus $4$

$4 x^{2} + (- 3 + 4) x - 3 = 0$

Step 2.1.3

Apply the distributive property.

$4 x^{2} - 3 x + 4 x - 3 = 0$

Step 2.2

Factor out the greatest common factor from each group.

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Step 2.2.1

Group the first two terms and the last two terms.

$(4 x^{2} - 3 x) + 4 x - 3 = 0$

Step 2.2.2

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

$x (4 x - 3) + 1 (4 x - 3) = 0$

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $4 x - 3$ .

$(4 x - 3) (x + 1) = 0$

Step 3

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

$4 x - 3 = 0$

$x + 1 = 0$

Step 4

Set $4 x - 3$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $4 x - 3$ equal to $0$ .

$4 x - 3 = 0$

Step 4.2

Solve $4 x - 3 = 0$ for $x$ .

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Step 4.2.1

Add $3$ to both sides of the equation.

$4 x = 3$

Step 4.2.2

Divide each term in $4 x = 3$ by $4$ and simplify.

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Step 4.2.2.1

Divide each term in $4 x = 3$ by $4$ .

$\frac{4 x}{4} = \frac{3}{4}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of $4$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{3}{4}$

Step 4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{4}$

Step 5

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

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Step 5.1

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

$x + 1 = 0$

Step 5.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 6

The final solution is all the values that make $(4 x - 3) (x + 1) = 0$ true.

$x = \frac{3}{4}, - 1$

Step 7

Find the values of $n$ that produce a value within the interval $(0, 9)$ .

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Step 7.1

The interval $(0, 9)$ does not contain $- 1$ . It is not part of the final solution.

$- 1$ is not on the interval

Step 7.2

The interval $(0, 9)$ contains $\frac{3}{4}$ .

$x = \frac{3}{4}$

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