The quadratic equation x^2 - 12x + 4 = 0 has solutions x_1 and x_2. What is the value of \… — SAT

The quadratic equation $x^2 - 12x + 4 = 0$ has solutions $x_1$ and $x_2$ . What is the value of $\frac{1}{x_1} + \frac{1}{x_2}$ ?

Answer: 3

The correct answer is 3.

By writing the expression

\frac{1}{x_1} + \frac{1}{x_2}

with a common denominator, we get

\frac{x_1 + x_2}{x_1 x_2}

. For the quadratic equation

x^2 - 12x + 4 = 0

, Vieta's formulas give the sum of the roots

x_1 + x_2 = 12

and the product of the roots

x_1 x_2 = 4

. Substituting these values into the fraction yields

\frac{12}{4} = 3

Find a common denominator to combine the terms in the given expression.

\frac{1}{x_1} + \frac{1}{x_2} = \frac{x_1 + x_2}{x_1 x_2}

To express the target quantity in terms of the sum and product of the quadratic solutions.

Apply Vieta's formulas to find the sum and product of the solutions from the quadratic equation

x^2 - 12x + 4 = 0

x_1 + x_2 = 12

and

x_1 x_2 = 4

For any quadratic equation

ax^2 + bx + c = 0

, the sum of the roots is

-\frac{b}{a}

and the product is

\frac{c}{a}

Substitute the sum and product values into the combined fraction.

\frac{12}{4} = 3

To compute the numerical value of the expression.

Sum and product of solutions of a quadratic equation

Question