Find the Roots (Zeros) f(x)=9x^2-54x-19

$f (x) = 9 x^{2} - 54 x - 19$

Step 1

Set $9 x^{2} - 54 x - 19$ equal to $0$ .

$9 x^{2} - 54 x - 19 = 0$

Step 2

Solve for $x$ .

Factor by grouping.

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 $a \cdot c = 9 \cdot - 19 = - 171$ and whose sum is $b = - 54$ .

Factor $- 54$ out of $- 54 x$ .

$9 x^{2} - 54 x - 19 = 0$

Rewrite $- 54$ as $3$ plus $- 57$

$9 x^{2} + (3 - 57) x - 19 = 0$

Apply the distributive property.

$9 x^{2} + 3 x - 57 x - 19 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(9 x^{2} + 3 x) - 57 x - 19 = 0$

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

$3 x (3 x + 1) - 19 (3 x + 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $3 x + 1$ .

$(3 x + 1) (3 x - 19) = 0$

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

$3 x + 1 = 0$

$3 x - 19 = 0$

Set $3 x + 1$ equal to $0$ and solve for $x$ .

Set $3 x + 1$ equal to $0$ .

$3 x + 1 = 0$

Solve $3 x + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$3 x = - 1$

Divide each term in $3 x = - 1$ by $3$ and simplify.

Divide each term in $3 x = - 1$ by $3$ .

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Set $3 x - 19$ equal to $0$ and solve for $x$ .

Set $3 x - 19$ equal to $0$ .

$3 x - 19 = 0$

Solve $3 x - 19 = 0$ for $x$ .

Add $19$ to both sides of the equation.

$3 x = 19$

Divide each term in $3 x = 19$ by $3$ and simplify.

Divide each term in $3 x = 19$ by $3$ .

$\frac{3 x}{3} = \frac{19}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{19}{3}$

Divide $x$ by $1$ .

$x = \frac{19}{3}$

The final solution is all the values that make $(3 x + 1) (3 x - 19) = 0$ true.

$x = - \frac{1}{3}, \frac{19}{3}$

Step 3

Find the Roots (Zeros) f(x)=9x^2-54x-19

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