Algebra Practice

Discriminant Practice Test

Calculate D=b²-4ac and use its sign and square status to classify quadratic roots and graph intersections.

Discriminant Practice Test

20 varied discriminant questions with worked explanations.

Instant feedback · Worked explanations

After the test · root classifier

The discriminant classifies quadratic roots before you finish solving the equation

For ax² + bx + c = 0, the discriminant D = b² − 4ac reveals whether the quadratic has two real roots, one repeated real root, or no real roots. When D is positive, one more check matters: if D is a perfect square, rational coefficients can lead to rational roots; if D is positive but not a perfect square, the roots are irrational. The same information also predicts how many times the parabola meets the x-axis.

signed coefficientsD = b² − 4acroot count x-interceptsperfect-square checkrepeated roots quadratic formula
Use the discriminant in two stages: classify by sign first, then—only when D is positive—check whether D is a perfect square.

Quick formula shelf

The essential formulas and classifications used on this page

Standard quadratic Read coefficients only from zero form ax² + bx + c = 0

a, b, and c include their signs; a must be nonzero.

Discriminant Root classifier D = b² − 4ac

The factor 4 and the signs of a, b, and c all matter.

Repeated root When D = 0 x = −b / (2a)

The ± part disappears because √D = 0.

Quadratic formula D sits inside the square root x = [−b ± √D] / (2a)

The discriminant controls what kind of square-root term appears.

Standard-form pipeline

Do not read coefficients until every term is on one side

Rearrange Move all terms to one side.

The target is ax² + bx + c = 0.

Identify Read a, b, and c with their signs.

Missing terms mean the corresponding coefficient is zero.

Calculate Substitute into D = b² − 4ac.

Square b itself and preserve negative values of a or c inside the product.

Classify Use the sign first, then square status if D > 0.

This determines real-root count and often rational-versus-irrational status.

Example
2x² − 7x − 4 = 0
a
a = 2
b
b = −7, not 7
c
c = −4, not 4
Missing term
For 3x² − 12 = 0, b = 0.

Sign → roots → graph

One discriminant sign gives three linked interpretations

D > 0

two distinct real roots
two x-intercepts

Then check whether D is a perfect square to distinguish rational from irrational roots for rational coefficients.

D = 0

one repeated real root
one x-intercept

The parabola touches the x-axis at its repeated root.

D < 0

no real roots
no real x-intercepts

The parabola does not meet the x-axis.

Visual graph connection

The discriminant predicts how the parabola interacts with the x-axis

D > 0 Two real x-intercepts.
D = 0 One repeated x-intercept.
D < 0 No real x-intercepts.

Positive D: square or nonsquare?

A positive discriminant gives two real roots, but not necessarily rational ones

Positive perfect square

D = 49
√D = 7

With rational coefficients, the quadratic formula produces rational roots.

Positive nonsquare

D = 12
√D = 2√3

The roots are real but irrational when the remaining radical does not simplify to a rational number.

D

D = b² − 4ac

This is the discriminant itself. Its sign is used for classification.

NOT
THE
SAME

√D

√D

This is the square-root term that appears inside the quadratic formula. For example, if D = 36, then √D = 6.

Repeated-root formula

When D = 0, the ± branches collapse to one value

Quadratic formula

x = [−b ± √0] / (2a)

The square-root term contributes zero.

BOTH
BRANCHES
COINCIDE

Repeated root

x = −b / (2a)

This single x-value is the repeated real root and the parabola's only x-intercept.

Connection to the quadratic formula

The discriminant is the part of the formula that controls the root type

This is why the test can ask for D, √D, root count, or the full roots as separate but connected skills.

x = [−b ± √(b² − 4ac)] / (2a)
Sign of D

Determines whether the square-root term is positive-real, zero, or nonreal.

Square status of D

For positive D, tells whether the radical simplifies to a rational value or remains irrational.

D = 0 shortcut

The formula reduces immediately to x = −b/(2a).

Common mistakes from the page

The test targets coefficient, sign, and interpretation errors around D

Using b instead of b²
D = b − 4ac. D = b² − 4ac.

The square on b is part of the definition.

Negative c loses its sign
For c = −4, substituting c = 4. Substitute the signed value: −4a(−4) changes the arithmetic.

Parentheses help preserve negative coefficients.

Factor 4 omitted
D = b² − ac. D = b² − 4ac.

The 4 is essential and can completely change the classification.

Coefficients read too early
Reading a, b, c before the equation is rearranged to equal zero. First rewrite into ax² + bx + c = 0.

Moving terms can change coefficient signs.

Every positive D called rational
D = 12 classified as producing rational roots. Positive D gives two real roots; check whether D is a perfect square.

A positive nonsquare discriminant produces irrational roots for rational coefficients.

D confused with √D
If D = 36, reporting D = 6. D = 36 and √D = 6.

The test explicitly distinguishes these two quantities.

Final discriminant checklist

Before selecting an answer, classify in the correct order

Standard form first Rewrite the equation as ax² + bx + c = 0 before reading coefficients.
Signed substitution Preserve zero and negative coefficients, square b, and include the factor 4.
Sign classification D > 0 gives two real roots, D = 0 one repeated root, and D < 0 no real roots.
Square-status check For positive D, perfect square versus nonsquare distinguishes rational from irrational roots for rational coefficients.