Solve by Factoring 3x^2-7x-5=0

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

Step 1

Use the quadratic formula to find the solutions.

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

Step 2

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

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

Step 3

Simplify.

Simplify the numerator.

Raise $- 7$ to the power of $2$ .

$x = \frac{7 \pm \sqrt{49 - 4 \cdot 3 \cdot - 5}}{2 \cdot 3}$

Multiply $- 4 \cdot 3 \cdot - 5$ .

Multiply $- 4$ by $3$ .

$x = \frac{7 \pm \sqrt{49 - 12 \cdot - 5}}{2 \cdot 3}$

Multiply $- 12$ by $- 5$ .

$x = \frac{7 \pm \sqrt{49 + 60}}{2 \cdot 3}$

Add $49$ and $60$ .

$x = \frac{7 \pm \sqrt{109}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{7 \pm \sqrt{109}}{6}$

Step 4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 7$ to the power of $2$ .

$x = \frac{7 \pm \sqrt{49 - 4 \cdot 3 \cdot - 5}}{2 \cdot 3}$

Multiply $- 4 \cdot 3 \cdot - 5$ .

Multiply $- 4$ by $3$ .

$x = \frac{7 \pm \sqrt{49 - 12 \cdot - 5}}{2 \cdot 3}$

Multiply $- 12$ by $- 5$ .

$x = \frac{7 \pm \sqrt{49 + 60}}{2 \cdot 3}$

Add $49$ and $60$ .

$x = \frac{7 \pm \sqrt{109}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{7 \pm \sqrt{109}}{6}$

Change the $\pm$ to $+$ .

$x = \frac{7 + \sqrt{109}}{6}$

Step 5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 7$ to the power of $2$ .

$x = \frac{7 \pm \sqrt{49 - 4 \cdot 3 \cdot - 5}}{2 \cdot 3}$

Multiply $- 4 \cdot 3 \cdot - 5$ .

Multiply $- 4$ by $3$ .

$x = \frac{7 \pm \sqrt{49 - 12 \cdot - 5}}{2 \cdot 3}$

Multiply $- 12$ by $- 5$ .

$x = \frac{7 \pm \sqrt{49 + 60}}{2 \cdot 3}$

Add $49$ and $60$ .

$x = \frac{7 \pm \sqrt{109}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{7 \pm \sqrt{109}}{6}$

Change the $\pm$ to $-$ .

$x = \frac{7 - \sqrt{109}}{6}$

Step 6

The final answer is the combination of both solutions.

$x = \frac{7 + \sqrt{109}}{6}, \frac{7 - \sqrt{109}}{6}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{7 + \sqrt{109}}{6}, \frac{7 - \sqrt{109}}{6}$

Decimal Form:

$x = 2.90671775 \dots, - 0.57338441 \dots$

Solve by Factoring 3x^2-7x-5=0

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