Quadratic Equation/Examples/3 x^2 + 24 x + 9 = 0

Example of Quadratic Equation

The quadratic equation:

$3 x^2 + 24 x + 9 = 0$

has the roots:

$x = -4 \pm \sqrt {13}$

Proof

\(\ds 3 x^2 + 24 x + 9\)

\(=\)

\(\ds 0\)

\(\ds \leadsto \ \ \)

\(\ds x^2 + 8 x + 3\)

\(=\)

\(\ds 0\)

dividing both sides by $3$

\(\ds \leadsto \ \ \)

\(\ds \paren {x + 4}^2 - 16 + 3\)

\(=\)

\(\ds 0\)

Completing the Square

\(\ds \leadsto \ \ \)

\(\ds \paren {x + 4}^2\)

\(=\)

\(\ds 13\)

simplifying

\(\ds \leadsto \ \ \)

\(\ds x + 4\)

\(=\)

\(\ds \pm \sqrt {13}\)

simplifying

The result follows.

$\blacksquare$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): completing the square
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): completing the square