Find the Sum and Product of the Roots of the Quadratic Equation 10x^2-2x+2=3x-4

$10 x^{2} - 2 x + 2 = 3 x - 4$

Find $a$ , $b$ , and $c$ .

Move all terms to the left side of the equation and simplify.

Move all the expressions to the left side of the equation.

Subtract $3 x$ from both sides of the equation.

$10 x^{2} - 2 x + 2 - 3 x = - 4$

Add $4$ to both sides of the equation.

$10 x^{2} - 2 x + 2 - 3 x + 4 = 0$

Simplify $10 x^{2} - 2 x + 2 - 3 x + 4$ .

Subtract $3 x$ from $- 2 x$ .

$10 x^{2} - 5 x + 2 + 4 = 0$

Add $2$ and $4$ .

$10 x^{2} - 5 x + 6 = 0$

Once the quadratic is in standard form, the values of $a$ , $b$ , and $c$ can be found.

$a x^{2} + b x + c$

Use the standard form of the equation to find $a$ , $b$ , and $c$ for this quadratic.

$a = 10$ , $b = - 5$ , $c = 6$

Substitute the values into the sum of roots formula $- \frac{b}{a}$ .

Cancel the common factor of $5$ and $10$ .

Factor $5$ out of $5$ .

$\frac{5 (1)}{10}$

Cancel the common factors.

Factor $5$ out of $10$ .

$\frac{5 \cdot 1}{5 \cdot 2}$

Cancel the common factor.

$\frac{5 \cdot 1}{5 \cdot 2}$

Rewrite the expression.

$\frac{1}{2}$

Substitute the values into the product of roots formula $\frac{c}{a}$ .

Cancel the common factor of $6$ and $10$ .

Factor $2$ out of $6$ .

$\frac{2 (3)}{10}$

Cancel the common factors.

Factor $2$ out of $10$ .

$\frac{2 \cdot 3}{2 \cdot 5}$

Cancel the common factor.

$\frac{2 \cdot 3}{2 \cdot 5}$

Rewrite the expression.

$\frac{3}{5}$

List the results.

Sum of the roots is $\frac{1}{2}$

Product of the roots is $\frac{3}{5}$