The Quadratic Formula, Step by Step (and Why You Should Memorise It)

The quadratic formula is one of the most useful things a 14-year-old learns. It will show up in physics, economics, computer science, and most engineering exams. Memorise it once, use it forever.

The formula

For any equation in the form ax² + bx + c = 0:

x = (-b ± √(b² - 4ac)) / 2a

That's it. Three coefficients in. Two solutions out.

Why ± gives two answers

The square root can be positive or negative, so you get two x-values. That matches the geometry — a parabola crosses the x-axis at most twice.

Worked example

Solve 2x² - 3x - 5 = 0

Step 1. Identify a, b, c.

• a = 2
• b = -3
• c = -5

Step 2. Compute the discriminant b² - 4ac.

• 9 - 4(2)(-5) = 9 + 40 = 49

Step 3. Take the square root.

• √49 = 7

Step 4. Apply the formula.

• x = (3 ± 7) / 4
• x = 10/4 or x = -4/4
• x = 2.5 or x = -1

Verify by substituting back. Both work.

Common pitfalls

• Wrong sign on b. It's minus b, so if b is negative, you get +. Easy to mess up.
• Forgetting to divide by 2a. Half the wrong answers come from this.
• Discriminant negative. Means there are no real solutions, just imaginary. Don't try to sqrt a negative.

When to use the quadratic formula vs. completing the square

• Use the formula when factoring is not obvious or impossible
• Use completing the square when you need the vertex form
• Use factoring when one factor is obvious (saves time in exams)

Memorise the formula. Get fast at it. It buys you marks across years of schooling.