Question

fill in the blank so that the resulting statement is true.

the solutions of a quadratic equation in standard form \\(ax^2 + bx + c = 0\\), \\(a \
eq 0\\), are given by the quadratic formula \\(x = \underline{\quad\quad\quad}\\)

the solutions of a quadratic equation in standard form \\(ax^2 + bx + c = 0\\), \\(a \
eq 0\\), are given by the quadratic formula \\(x = \\)

options:

1. \\(\frac{-b - \sqrt{b^2 - ac}}{2a}\\)
2. \\(\frac{-b + \sqrt{b^2 - 4ac}}{a}\\)
3. \\(\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\\)
4. \\(\frac{-b \pm \sqrt{b^2 - ac}}{2a}\\)

Explanation:

Identify the standard form

Using the Quadratic Equations knowledge point

$$ ax^2 + bx + c = 0, \quad a eq 0 $$

State the quadratic formula

The standard formula derived by completing the square is:

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

Match with the given options

Comparing the standard formula to the dropdown choices:

• Option 1: \(\frac{-b - \sqrt{b^2 - ac}}{2a}\) (Incorrect)
• Option 2: \(\frac{-b + \sqrt{b^2 - 4ac}}{a}\) (Incorrect)
• Option 3: \(\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) (Correct)
• Option 4: \(\frac{-b \pm \sqrt{b^2 - ac}}{2a}\) (Incorrect)

Answer:

The solutions of a quadratic equation in standard form \(ax^2 + bx + c = 0\), \(a
eq 0\), are given by the quadratic formula \(x =\) <blank>\(\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)</blank>.