Algebra Practice

Quadratic Equations Practice Test

Choose and apply efficient methods for factorable, irrational, repeated, and nonreal quadratic solutions.

Quadratic Equations Practice Test

This test has 20 questions

Instant feedback · Worked explanations

After the test · quadratic method lab

Quadratic equations are not about memorizing one procedure — they are about recognizing the fastest valid method

This mixed review combines factoring, the zero-product property, the square-root property, completing the square, the quadratic formula, radical simplification, and discriminants. The equations may have rational, irrational, repeated, or nonreal solutions, so the first decision is often not “what is x?” but “which method fits the structure of this quadratic?”

factoringzero-product propertysquare roots completing the squarequadratic formula radical simplificationdiscriminant
A strong solver checks structure first: common factor, perfect-square form, factorability, and discriminant can all make the solution shorter.

Method selector

Choose the method from the equation's structure

Factoring
The quadratic factors cleanly, especially after removing a greatest common factor.
Best when integer or simple rational factors are visible.
Square-root property
A squared expression can be isolated: u² = k.
Remember both branches when k > 0: u = ±√k.
Completing the square
The equation is easy to normalize and convert into a perfect-square trinomial.
Useful when factoring is awkward but exact structure matters.
Quadratic formula
Any quadratic in standard form ax² + bx + c = 0.
The universal method; especially useful for irrational or nonreal roots.

Quick formula shelf

Core identities and formulas for the methods named on this page

Standard form Quadratic equation ax² + bx + c = 0, with a ≠ 0

Most classification tools assume the equation has first been written in this form.

Quadratic formula Universal solution formula x = (−b ± √(b² − 4ac)) / (2a)

Keep the denominator 2a grouped under the entire numerator.

Discriminant Classify roots before solving Δ = b² − 4ac

Its sign determines whether the quadratic has two, one, or no real roots.

Square-root property When a square is isolated u² = k → u = ±√k

The ± is essential when k is positive.

Zero-product property After factoring AB = 0 → A = 0 or B = 0

This is what turns a factored quadratic into separate linear equations.

Completing the square Perfect-square pattern x² + bx + (b/2)² = (x + b/2)²

After normalizing the x² coefficient to 1, add the same quantity to both sides.

Discriminant classification

The sign of b² − 4ac predicts the number of real solutions

Δ > 0

two distinct real roots

If the discriminant is not a perfect square, the roots are typically irrational.

Δ = 0

one repeated real root

The parabola touches the x-axis at exactly one point.

Δ < 0

no real roots

The square root in the quadratic formula is taken from a negative value, producing nonreal solutions.

Visual root picture

Real solutions are the x-coordinates where the parabola meets the x-axis

Two real roots The parabola crosses the x-axis twice.
Repeated real root The parabola touches the x-axis once.
No real roots The parabola does not meet the x-axis.
Factoring route · apply the page's “common factor first” rule
START
2x² − 10x = 0 Both terms share a factor of 2x.
GCF
2x(x − 5) = 0 Factor out the common factor before doing anything more complicated.
ZERO PRODUCT
2x = 0 or x − 5 = 0 Each factor can produce a root.
ROOTS
x = 0 or x = 5 Do not omit zero after factoring out x.

Square-root property

A positive isolated square has two branches

Isolate the square

(x − 3)² = 16

The left side is already a perfect square.

TAKE
±√

Keep both branches

x − 3 = ±4
x = 7 or x = −1

Dropping the negative branch loses a valid solution.

Completing the square

Build a perfect square, then use the square-root property

1 · Normalize x² + 6x = 7

The x² coefficient is already 1.

2 · Half b 6/2 = 3

Square it: 3² = 9.

3 · Add both sides x² + 6x + 9 = 16

Keep the equation balanced.

4 · Rewrite (x + 3)² = 16

Now apply ±√16.

Quadratic formula anatomy

Keep the entire numerator over 2a

The page specifically warns against losing the grouping in the denominator. Treat the numerator −b ± √(b² − 4ac) as one complete expression.

x = [−b ± √(b² − 4ac)] / (2a)
−b

Use the opposite of the actual b-value, including its sign.

√(b² − 4ac)

This contains the discriminant and controls the root type.

2a

The whole numerator is divided by 2a, not only the radical term.

Root-type review

The test mixes several kinds of quadratic solutions

Rational roots

Often appear in cleanly factorable equations or when the discriminant is a perfect square.

Irrational roots

Often remain in radical form after the quadratic formula or square-root property.

Repeated root

Occurs when the discriminant equals zero; both branches collapse to the same value.

Nonreal roots

Occur when the discriminant is negative, so there are no real x-intercepts.

Common mistakes from the page

The test targets several errors that can remove or reverse valid roots

Sign copied from factor
(x − 4) = 0 reported as x = −4. Solve the linear factor: x = 4.

The sign inside a factor is not copied directly as the root.

Zero root omitted
x(x − 5) = 0 reported only as x = 5. x = 0 or x = 5.

Every factor must be set equal to zero.

Missing ±
(x − 2)² = 9 → x − 2 = 3 only. x − 2 = ±3.

A positive square-root equation has two branches unless they merge into one repeated root.

Formula denominator split
Only the radical term is divided by 2a. The entire numerator −b ± √Δ is divided by 2a.

Use parentheses or a fraction bar to preserve the formula's structure.

Negative discriminant misread
Δ < 0 interpreted as “zero solutions.” It means no real roots; the quadratic has nonreal solutions.

The page distinguishes real-root classification from the broader existence of complex roots.

Final quadratic checklist

Before solving, classify the structure; after solving, classify the roots

Common factor first Factor out the GCF before choosing a more complicated method.
Preserve both branches Use ± when taking the square root of a positive isolated square.
Protect formula structure Keep −b ± √Δ together over the full denominator 2a.
Read the discriminant Positive, zero, or negative discriminant predicts two, one repeated, or no real roots.