Find the Roots (Zeros) g(x)=5x-12x^2+3

$g (x) = 5 x - 12 x^{2} + 3$

Set $5 x - 12 x^{2} + 3$ equal to $0$ .

$5 x - 12 x^{2} + 3 = 0$

Solve for $x$ .

Factor the left side of the equation.

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$5 u - 12 u^{2} + 3 = 0$

Factor by grouping.

Reorder terms.

$- 12 u^{2} + 5 u + 3 = 0$

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 $a \cdot c = - 12 \cdot 3 = - 36$ and whose sum is $b = 5$ .

Factor $5$ out of $5 u$ .

$- 12 u^{2} + 5 (u) + 3 = 0$

Rewrite $5$ as $- 4$ plus $9$

$- 12 u^{2} + (- 4 + 9) u + 3 = 0$

Apply the distributive property.

$- 12 u^{2} - 4 u + 9 u + 3 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(- 12 u^{2} - 4 u) + 9 u + 3 = 0$

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

$4 u (- 3 u - 1) - 3 (- 3 u - 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $- 3 u - 1$ .

$(- 3 u - 1) (4 u - 3) = 0$

Replace all occurrences of $u$ with $x$ .

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

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

$- 3 x - 1 = 0$

$4 x - 3 = 0$

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 $x$ .

Add $1$ to 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 $4 x - 3$ equal to $0$ and solve for $x$ .

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

$4 x - 3 = 0$

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

Add $3$ to both sides of the equation.

$4 x = 3$

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

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

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

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

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

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

Find the Roots (Zeros) g(x)=5x-12x^2+3

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