Find the Roots (Zeros) f(x)=3x^2-5x-7

$f (x) = 3 x^{2} - 5 x - 7$

Step 1

Set $3 x^{2} - 5 x - 7$ equal to $0$ .

$3 x^{2} - 5 x - 7 = 0$

Step 2

Solve for $x$ .

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 3$ , $b = - 5$ , and $c = - 7$ into the quadratic formula and solve for $x$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (3 \cdot - 7)}}{2 \cdot 3}$

Simplify.

Simplify the numerator.

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 3 \cdot - 7}}{2 \cdot 3}$

Multiply $- 4 \cdot 3 \cdot - 7$ .

Multiply $- 4$ by $3$ .

$x = \frac{5 \pm \sqrt{25 - 12 \cdot - 7}}{2 \cdot 3}$

Multiply $- 12$ by $- 7$ .

$x = \frac{5 \pm \sqrt{25 + 84}}{2 \cdot 3}$

Add $25$ and $84$ .

$x = \frac{5 \pm \sqrt{109}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{5 \pm \sqrt{109}}{6}$

The final answer is the combination of both solutions.

$x = \frac{5 + \sqrt{109}}{6}, \frac{5 - \sqrt{109}}{6}$

$x = \frac{5 \pm \sqrt{109}}{6}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5 \pm \sqrt{109}}{6}$

Decimal Form:

$x = 2.57338441 \dots, - 0.90671775 \dots$

Step 4