Question

practice assignment 5.1 quadratic functions

question 9

use the quadratic formula to solve the equation (9x^2 + 18x + 10 = 0).

enter multiple answers as a list separated by commas. example: (2 + 2i, 2 - 2i)

Explanation:

⚡ Using what you learned: quadratic formula and its applications

Step 1: Identify Coefficients

Identify the coefficients \(a\), \(b\), and \(c\) from the standard form equation \(ax^2 + bx + c = 0\):

$$ 9x^2 + 18x + 10 = 0 $$

$$ a = 9, \quad b = 18, \quad c = 10 $$

Step 2: Apply the Quadratic Formula

Substitute the coefficients into the quadratic formula:

$$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$

$$ x = \frac{-18 \pm \sqrt{18^2 - 4(9)(10)}}{2(9)} $$

Step 3: Simplify the Discriminant

Calculate the value under the square root:

$$ 18^2 = 324 $$

$$ 4(9)(10) = 360 $$

$$ b^2 - 4ac = 324 - 360 = -36 $$

Step 4: Simplify the Radical with Complex Numbers

Simplify the square root of the negative number:

$$ \sqrt{-36} = \sqrt{36} \cdot \sqrt{-1} = 6i $$

Step 5: Solve for x

Substitute back into the expression and simplify:

$$ x = \frac{-18 \pm 6i}{18} $$

$$ x = \frac{-18}{18} \pm \frac{6i}{18} $$

$$ x = -1 \pm \frac{1}{3}i $$

Answer:

-1 + 1/3i, -1 - 1/3i