Quadratic Equations with No Real Solution Practice Test

After the test · real-root exclusion lab

A negative discriminant means the quadratic never reaches zero at a real x-value, but the equation still has two complex roots

Every question on this page centers on D < 0. The same situation can be recognized from the discriminant, from a graph with no x-intercepts, from vertex form showing a positive minimum or negative maximum, or from the quadratic formula producing a square root of a negative number. Parameter questions ask for the strict ranges that keep D negative, with D = 0 endpoints excluded.

D < 0no real x-interceptscomplex conjugates parameter inequalitiesvertex formminimum / maximum completing the square

“No real solution” is not the same as “no solution.” For a real-coefficient quadratic with D < 0, the two roots are complex conjugates.

Quick formula shelf

Core relationships for the no-real-root case

Discriminant condition No real roots D = b² − 4ac < 0

The inequality is strict; D = 0 produces a repeated real root.

Quadratic formula Complex roots x = [−b ± √D] / (2a)

When D is negative, rewrite the square root using i.

Imaginary unit Convert a negative radical √(−m) = i√m, for m > 0

This keeps the solution process going in the complex number system.

Vertex form Graph stays away from y = 0 y = a(x − h)² + k

If a > 0 and k > 0, the minimum is positive; if a < 0 and k < 0, the maximum is negative.

Visual real-axis exclusion zone

No real x-intercepts means the entire parabola stays on one side of the x-axis

Upward parabola Positive minimum → the graph stays above the x-axis.

Downward parabola Negative maximum → the graph stays below the x-axis.

D < 0 → complex roots

A negative discriminant changes the number system, not the existence of solutions

1 · Discriminant D = −20

There are no real roots because D is negative.

2 · Radical √D = √(−20)

The square-root term is not real.

3 · Introduce i √(−20) = i√20 = 2i√5

Simplify the positive radical normally.

4 · Finish formula x = [−b ± 2i√5] / (2a)

The ± branches form a complex conjugate pair.

Complex conjugate pair

Real-coefficient quadratics produce complex roots in matching ± imaginary pairs

First root

x₁ = p + qi

The real part is p and the imaginary part is +q.

SAME REAL PART
OPPOSITE i SIGN

Conjugate root

x₂ = p − qi

The real part stays the same while the imaginary sign reverses.

Parameter ranges

Require the discriminant to stay strictly negative

1 · Build D D(parameter) = b² − 4ac

Write the discriminant in terms of the parameter.

2 · Require no real roots D(parameter) < 0

Use a strict inequality.

3 · Solve the inequality Find the interval or intervals where D is negative.

The sign of the resulting expression determines the valid region.

4 · Exclude endpoints D = 0 values stay out.

Those values create repeated real roots, not no-real-root cases.

D = 0

Open circles represent excluded endpoint values where the discriminant becomes zero.

Completing the square / vertex-form proof

START

y = x² + 6x + 13 Completing the square reveals the minimum directly.

COMPLETE

y = (x + 3)² + 4 The squared term is always at least 0 over the reals.

MINIMUM

minimum y = 4 The graph never reaches y = 0.

CONCLUDE

No real x-intercepts The equation x² + 6x + 13 = 0 therefore has no real roots.

Minimum and maximum signals

Vertex form can prove a quadratic never reaches zero without calculating D first

This matches the page's use of graph position, extrema, and completing the square.

Positive minimum y = 2(x − 1)² + 5

The minimum is 5, so the graph is always above the x-axis and has no real zeros.

Negative maximum y = −3(x + 2)² − 4

The maximum is −4, so the graph is always below the x-axis and has no real zeros.

Use precise solution language

“No real roots” is correct; “no solutions at all” is not

Incomplete statement

Saying only that the equation has “no solutions” ignores the complex number system.

D < 0 → no real roots

Complete interpretation

A real quadratic with negative discriminant has no real roots but does have two complex conjugate roots.

p + qi and p − qi

Common mistakes from the page

The main errors are stopping too early or allowing forbidden D = 0 endpoints

“No solutions” claimed

D < 0 described as meaning the equation has no solutions of any kind. It has no real roots but two complex conjugate roots.

The quadratic formula continues using i.

D = 0 endpoint included

Using D ≤ 0 for a no-real-root parameter range. Use D < 0.

D = 0 gives one repeated real root and must be excluded.

Graph position misread

An upward parabola with positive minimum treated as having real zeros. A positive minimum keeps the entire graph above the x-axis.

No crossing means no real x-intercepts.

Complex pair not conjugate

Changing both the real and imaginary parts between the two roots. The real part matches; only the sign of the imaginary part changes.

Real-coefficient quadratics produce conjugate complex roots.

Final no-real-solution checklist

Before selecting an answer, confirm algebra, graph, and parameter conditions agree

D is strictly negative The discriminant satisfies D < 0, not D ≤ 0.

No real x-intercepts The parabola stays entirely above or entirely below the x-axis.

Complex pair completed The negative radical is rewritten with i and both conjugate roots are reported.

Endpoints excluded Any parameter value making D = 0 is removed from a no-real-root range.