Quadratic Equations with One Solution Practice Test

After the test · repeated-root lab

One distinct quadratic solution appears when two roots collapse into the same value

This practice page isolates the repeated-root case and connects four equivalent views: discriminant zero, a perfect-square quadratic, a vertex on the x-axis, and a parabola tangent to that axis. Parameter questions ask for the exact equality that forces the two quadratic-formula branches to coincide.

repeated rootsD = 0perfect squares tangencyvertex formparameter equality

One solution means one distinct real x-value with multiplicity two — not two different real solutions.

Four equivalent signals

The same repeated-root case appears in algebra, factoring, graphing, and the quadratic formula

Discriminant

D = 0. The square-root term in the quadratic formula vanishes.

Factored form

The same linear factor appears twice: (x − r)² = 0.

Vertex form

The vertex lies on y = 0, so the parabola's extreme point sits exactly on the x-axis.

Graph

The parabola is tangent to the x-axis: it touches once instead of crossing through.

Quick formula shelf

Core relationships for one-solution quadratics

Repeated-root condition Zero discriminant D = b² − 4ac = 0

This is an equality, not an interval.

Repeated-root formula Quadratic formula collapses x = −b / (2a)

When D = 0, √D = 0 and both ± branches give the same value.

Perfect square Repeated factor (x − r)² = 0 → x = r

The root has multiplicity two but only one distinct value.

Vertex form Tangent parabola y = a(x − h)²

The vertex is (h, 0), exactly on the x-axis.

Visual tangency

A repeated root is the x-coordinate where the parabola touches the axis at its vertex

tangent vertex

One x-intercept

The graph has one distinct point where y = 0.

Vertex on the axis

The tangent point is also the vertex because the extreme value is exactly 0.

No crossing

The graph touches and turns rather than passing through the axis.

Perfect-square route

A repeated factor produces one distinct root even though the factor occurs twice

1 · Quadratic x² − 8x + 16 = 0

This trinomial is a perfect square.

2 · Factor (x − 4)² = 0

The same factor occurs twice.

3 · Set factor to zero x − 4 = 0

There is only one factor equation to solve.

4 · Distinct solution x = 4

Multiplicity two, one distinct real solution.

Repeated root versus two roots

D > 0

Two distinct real roots The ± branches differ because √D is positive.

D = 0

One repeated real root √D = 0, so adding or subtracting zero gives the same result.

MULTIPLICITY

The root counts twice algebraically But it is still only one distinct solution value.

Parameter conditions

For exactly one real solution, solve the equality that forces D to zero

1 · Standard form Identify a, b, c in terms of the parameter.

Preserve all signs.

2 · Build discriminant D(parameter) = b² − 4ac

The parameter may appear in one or several coefficients.

3 · Set equality D(parameter) = 0

Do not use >, <, ≥, or ≤ when the goal is exactly one repeated root.

4 · Solve Find the exact parameter value or values.

These are the values at which the two formula branches coincide.

The one-solution condition is a boundary equality. It is not a range of nearby parameter values.

Why ± does not create two roots when D = 0

Both quadratic-formula branches become identical

Plus branch

x₁ = [−b + √0] / (2a)
x₁ = −b / (2a)

Adding zero changes nothing.

SAME
VALUE

Minus branch

x₂ = [−b − √0] / (2a)
x₂ = −b / (2a)

Subtracting zero gives the same result.

Vertex-form signal

A vertex on y = 0 is the geometric signature of one repeated root

This directly connects the page's vertex-form, tangency, and repeated-root skills.

Upward opening y = 2(x − 3)²

The minimum occurs at (3, 0), so the only x-intercept is x = 3.

Downward opening y = −4(x + 1)²

The maximum occurs at (−1, 0), so the only x-intercept is x = −1.

Common condition k = 0 in y = a(x − h)² + k

Regardless of opening direction, vertex y-coordinate 0 creates tangency to the x-axis.

Factor-sign check

Factor signs reverse when converted into root values

Repeated factor

(x + 5)² = 0

The factor contains +5.

SET
FACTOR
= 0

Repeated root

x + 5 = 0
x = −5

Do not copy the sign from the factor directly.

Common mistakes from the page

The main errors come from treating a repeated root as if it behaved like two distinct roots

Writing ± when square = 0

(x − 4)² = 0 written as x − 4 = ±0 and then treated as two answers. x − 4 = 0, so x = 4.

+0 and −0 are the same number.

Repeated root counted twice

(x − 3)² = 0 described as two distinct real solutions. There is one distinct real solution with multiplicity two.

Multiplicity and number of distinct solutions are different ideas.

Parameter interval used

Using D ≥ 0 or a range when exactly one solution is required. Solve D = 0 exactly.

The repeated-root condition is an equality.

Factor sign copied

(x + 2)² = 0 reported as x = 2. x + 2 = 0 gives x = −2.

Factor signs reverse when solving for the root.

Final one-solution checklist

Before selecting an answer, confirm all repeated-root signals agree

D = 0 The discriminant is exactly zero, not merely nonnegative.

Repeated factor The quadratic can be represented with the same linear factor twice.

Tangent vertex The parabola touches the x-axis at its vertex and has one distinct x-intercept.

One distinct value The repeated root has multiplicity two but must not be counted as two distinct solutions.