The Quadratic Formula

The quadratic formula

The quadratic formula solves any equation of the form $ax^2 + bx + c = 0$:

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

It always works — even when factoring doesn't. The $\pm$ means there are (potentially) two solutions.

Step-by-step solution: Solve $2x^2 - 5x - 3 = 0$

Step 1 — Identify $a$, $b$, $c$. $a = 2$, $b = -5$, $c = -3$.

Step 2 — Compute the discriminant $\Delta = b^2 - 4ac$:

$\Delta = (-5)^2 - 4(2)(-3) = 25 + 24 = 49$

Since $\Delta > 0$, there are two distinct real roots.

Step 3 — Apply the formula:

$x = \frac{-(-5) \pm \sqrt{49}}{2(2)} = \frac{5 \pm 7}{4}$

Step 4 — Compute both roots:

$x_1 = \frac{5 + 7}{4} = \frac{12}{4} = 3$

$x_2 = \frac{5 - 7}{4} = \frac{-2}{4} = -\frac{1}{2}$

Step 5 — Check both roots.

• $x = 3$: $2(9) - 5(3) - 3 = 18 - 15 - 3 = 0$ ✓
• $x = -1/2$: $2(1/4) - 5(-1/2) - 3 = 1/2 + 5/2 - 3 = 0$ ✓

The discriminant tells you what to expect

• $\Delta > 0$: two real roots (parabola crosses x-axis twice)
• $\Delta = 0$: one repeated root (parabola touches x-axis)
• $\Delta < 0$: no real roots (parabola misses x-axis)

Common mistakes

• Sign errors with $-b$. If $b = -5$, then $-b = -(-5) = +5$. Double-check the sign.
• Forgetting $2a$ in the denominator. The formula divides by $2a$, not just $a$ or $2$.
• Arithmetic errors in $b^2 - 4ac$. Especially when $c$ is negative: $-4(2)(-3) = +24$, not $-24$.

Try it in the visualization

Adjust $a$, $b$, $c$ and watch the parabola shift. The roots (if real) appear as green dots on the x-axis. The discriminant value and its sign are shown — change the sliders to see all three cases (two roots, one root, no real roots).