Quadratic Formula Practice Test | Algebra Practice Test

After the test · quadratic formula cockpit

Using the quadratic formula correctly is a sequence of sign, grouping, discriminant, radical, and branch decisions

This practice set focuses on every stage of the formula: identifying signed coefficients, computing the discriminant, preserving the full denominator 2a, simplifying radicals, and evaluating both ± branches. The answer may be rational, irrational, repeated, or complex, so the discriminant does not merely sit inside the radical — it predicts what kind of roots the formula will produce.

signed coefficientsdiscriminant± branches denominator groupingradical simplification complex numbersexact roots

The formula itself is reliable; most mistakes come from feeding it the wrong signed coefficients or breaking its grouping during substitution.

Formula anatomy

Treat the quadratic formula as four connected parts

x = [−b ± √(b² − 4ac)] / (2a)

−b

Use the opposite of the actual signed b-value.

b² − 4ac

This is the discriminant. Its sign controls the root type.

±

This creates two calculations unless both branches collapse into one repeated root.

2a

The entire numerator is divided by the entire quantity 2a.

Equation

3x² − 5x − 2 = 0

a

a = 3

b

b = −5 — the minus sign belongs to the coefficient.

c

c = −2 — again, include the sign.

Signed-b checkpoint

When b is negative, the outer minus in −b changes it again

Coefficient read from equation

3x² − 5x − 2 = 0

b = −5

The signed coefficient is negative five.

APPLY
−b

Formula substitution

−b = −(−5) = 5

Do not drop the formula's minus or silently replace b by its absolute value.

Discriminant gate

Compute Δ = b² − 4ac before simplifying the full formula

Δ > 0

two distinct real roots

If Δ is a perfect square, the roots may be rational; otherwise they are often irrational.

Δ = 0

one repeated real root

The ± branches produce the same value because √0 = 0.

Δ < 0

two complex conjugate roots

The calculation continues using i; a negative discriminant does not end the solution process.

Two formula branches

The ± symbol means evaluate the numerator twice

Plus branch

x₁ = [−b + √Δ] / (2a)

Use the positive radical contribution.

±

Minus branch

x₂ = [−b − √Δ] / (2a)

Repeat the same denominator and coefficients, changing only the branch sign.

Denominator grouping · one of the page's key warnings

Correct grouping x = [−b ± √(b² − 4ac)] / (2a)

Incorrect structure x = −b ± [√(b² − 4ac) / (2a)]

Radical simplification

Keep exact roots and simplify the square root before reducing the final expression

1 · Discriminant Δ = 48

The radical is √48.

2 · Factor 48 = 16·3

Find a perfect-square factor.

3 · Simplify √48 = 4√3

Extract √16 = 4.

4 · Substitute x = [−b ± 4√3] / (2a)

Then reduce any common numerical factor if valid.

Negative discriminant → complex roots

A negative discriminant introduces i instead of stopping the calculation

This is one of the page's explicit key ideas.

Negative radical

√(−20)

Separate the negative factor from the positive magnitude.

Introduce i

√(−20) = i√20

Use √(−1) = i.

Simplify

i√20 = 2i√5

The two formula branches then produce a complex conjugate pair.

Root types in this practice set

The formula handles rational, irrational, repeated, and complex answers in one framework

Rational

Often occurs when the discriminant is a nonnegative perfect square and the resulting fraction simplifies cleanly.

Irrational

Occurs when the positive discriminant is not a perfect square and the radical remains in exact form.

Repeated

Occurs when Δ = 0, so both ± branches collapse to the same root.

Complex

Occurs when Δ < 0 and the radical introduces i.

Exact roots

Do not replace an exact radical with an unnecessary decimal unless the problem asks for approximation

Exact form

x = (3 ± √5) / 2

This preserves the exact value of both irrational roots.

Approximation

x ≈ 2.618 or 0.382

Useful only when a decimal approximation is specifically needed; otherwise exact form is preferable.

Common mistakes from the page

The test directly targets sign and grouping failures inside the formula

Minus before b dropped

Using b instead of −b in the numerator. Write −(b) during substitution, especially when b is negative.

This preserves the actual formula structure.

Discriminant sign changed

b² − 4ac rewritten as b² + 4ac. The discriminant is always b² − 4ac.

Any signs belonging to a or c are already contained in their substituted values.

Only radical divided

−b left outside while only √Δ is divided by 2a. The complete numerator −b ± √Δ is over 2a.

Use brackets or a fraction bar to protect the grouping.

One branch omitted

Evaluating only the + branch. Evaluate both + and − unless Δ = 0 causes the branches to coincide.

The ± symbol represents two calculations.

Negative Δ treated as stop

Declaring “no solutions” when Δ < 0. There are no real roots, but the formula continues into complex roots using i.

This distinction is explicitly emphasized in the page's key ideas.

Final quadratic-formula checklist

Before choosing an answer, audit the formula from coefficients to both final roots

Signed coefficients a, b, and c are read from standard form with their signs included.

Correct discriminant Δ is calculated as b² − 4ac, with all substituted signs preserved.

Grouping + both branches The complete numerator is over 2a and both ± branches are evaluated.

Exact simplification Radicals are simplified, repeated roots recognized, and negative discriminants converted into complex roots using i.