Finite Math Examples

$4 x^{2} - 17 x - 15$

Step 1

Set $4 x^{2} - 17 x - 15$ equal to $0$ .

$4 x^{2} - 17 x - 15 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor by grouping.

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Step 2.1.1

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 $a \cdot c = 4 \cdot - 15 = - 60$ and whose sum is $b = - 17$ .

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Step 2.1.1.1

Factor $- 17$ out of $- 17 x$ .

$4 x^{2} - 17 x - 15 = 0$

Step 2.1.1.2

Rewrite $- 17$ as $3$ plus $- 20$

$4 x^{2} + (3 - 20) x - 15 = 0$

Step 2.1.1.3

Apply the distributive property.

$4 x^{2} + 3 x - 20 x - 15 = 0$

Step 2.1.2

Factor out the greatest common factor from each group.

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Step 2.1.2.1

Group the first two terms and the last two terms.

$(4 x^{2} + 3 x) - 20 x - 15 = 0$

Step 2.1.2.2

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

$x (4 x + 3) - 5 (4 x + 3) = 0$

Step 2.1.3

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

$(4 x + 3) (x - 5) = 0$

Step 2.2

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

$4 x + 3 = 0$

$x - 5 = 0$

Step 2.3

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

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Step 2.3.1

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

$4 x + 3 = 0$

Step 2.3.2

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

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Step 2.3.2.1

Subtract $3$ from both sides of the equation.

$4 x = - 3$

Step 2.3.2.2

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

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Step 2.3.2.2.1

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

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

Step 2.3.2.2.2

Simplify the left side.

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Step 2.3.2.2.2.1

Cancel the common factor of $4$ .

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Step 2.3.2.2.2.1.1

Cancel the common factor.

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

Step 2.3.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 2.3.2.2.3

Simplify the right side.

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Step 2.3.2.2.3.1

Move the negative in front of the fraction.

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

Step 2.4

Set $x - 5$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 2.4.2

Add $5$ to both sides of the equation.

$x = 5$

Step 2.5

The final solution is all the values that make $(4 x + 3) (x - 5) = 0$ true.

$x = - \frac{3}{4}, 5$

Step 3