Quadratic Formula Examples: 8 Worked Problems

Starting simple: rational roots

The first two quadratic formula examples have perfect-square discriminants, so the roots come out as whole numbers or simple fractions.

1. x² − 5x + 6 = 0worked example

a = 1, b = −5, c = 6read off the coefficients
25 − 24 = 1the discriminant, a perfect square
x = (5 ± 1) ÷ 2−b is +5 because b was negative
x = 3 or 2both roots are whole numbers

2. 2x² + 7x + 3 = 0worked example

49 − 24 = 25perfect square again
x = (−7 ± 5) ÷ 42a = 4
x = −½ or −3one root is a fraction

Irrational roots and surds

When the discriminant is positive but not a perfect square, the exact answer contains a surd and the decimals are only an approximation.

3. x² − x − 1 = 0worked example

1 − (−4) = 5careful: 4ac is negative here, so it is added
x = (1 ± √5) ÷ 2√5 does not simplify
≈ 1.618 or −0.618the positive root is the golden ratio

4. 2x² + 4x − 3 = 0, with simplifyingworked example

16 + 24 = 40the discriminant
x = (−4 ± √40) ÷ 4substitute
x = (−4 ± 2√10) ÷ 4√40 = 2√10
x = (−2 ± √10) ÷ 2every term divides by 2

Repeated and complex roots

5. 4x² − 12x + 9 = 0worked example

144 − 144 = 0zero discriminant
x = 12 ÷ 8 = 1.5the ± adds nothing
one repeated rootthe quadratic is (2x − 3)²

6. x² + 2x + 5 = 0worked example

4 − 20 = −16negative discriminant
x = (−2 ± 4i) ÷ 2√−16 = 4i
x = −1 ± 2ia complex conjugate pair

Rearranging first, and a word problem

The last two need work before the formula can be applied at all — which is where marks are most often lost.

7. 3x² = 5x + 2worked example

3x² − 5x − 2 = 0move everything to one side first
25 + 24 = 49the discriminant, a perfect square
x = (5 ± 7) ÷ 6substitute
x = 2 or −⅓two rational roots

8. A rectangle 3 m longer than it is wide, area 40 m²worked example

w(w + 3) = 40width times length equals the area
w² + 3w − 40 = 0expand and set to zero
9 + 160 = 169 = 13²the discriminant
w = 5 or −8−8 is rejected — a width cannot be negative

That last line is the part that separates a maths answer from a correct one. The formula returns both roots faithfully; deciding which of them the question can actually accept is your job, not the formula's.

Checking every one of them

Each answer above can be checked in seconds without re-solving, using the sum and product of the roots.

Example	Roots	Sum = −b/a	Product = c/a
x² − 5x + 6	3 and 2	5 ✓	6 ✓
2x² + 7x + 3	−½ and −3	−3.5 ✓	1.5 ✓
3x² − 5x − 2	2 and −⅓	5/3 ✓	−2/3 ✓

Questions about quadratic formula examples

Do I always have to rearrange into standard form?

Yes. The formula reads a, b and c from ax² + bx + c = 0, so everything must be on one side with zero on the other before you start.

What if one root does not make sense in the question?

Discard it and say why. A negative length, a negative time or a fractional number of people are all valid roots of the equation and invalid answers to the problem.

Should I give the surd or the decimal?

Give the exact surd unless a decimal is asked for. If both are useful, show the exact form and then its decimal value.

How do I check my two roots quickly?

Add them and multiply them. The sum must equal −b/a and the product must equal c/a.