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Quizzes > High School Quizzes > Electives

Polynomial Root Multiplicity 2 Practice Quiz

Practice solving roots with interactive challenge questions

Difficulty: Moderate
Grade: Grade 9
Study OutcomesCheat Sheet
Colorful paper art promoting a Double Trouble Roots algebra practice quiz for high school students.

In quadratic equations, a root that occurs twice is known as what?
Irrational root
Single root
Double root
Complex root
A repeated root is called a double root because it appears twice in the factorization of the equation. This repeated occurrence is a defining feature when the discriminant is zero.
What is the necessary condition for a quadratic equation ax² + bx + c = 0 to have a repeated root?
b² < 4ac
b² = 4ac
a + b + c = 0
b² > 4ac
A quadratic equation has a repeated root when its discriminant is zero, meaning b² - 4ac must equal 0. This condition ensures both roots coincide.
If a quadratic equation has a repeated root, what is the value of its discriminant (b² - 4ac)?
Negative
0
Positive
Undefined
The discriminant being zero is the key feature of equations with repeated roots. This guarantees that both solutions are equal.
Which form represents a quadratic equation with a repeated root r?
a(x - 2r)² = 0
a(x - r) = 0
a(x - r)(x + r) = 0
a(x - r)² = 0
When a quadratic has a repeated root r, it factors perfectly as a(x - r)² = 0. This demonstrates the double occurrence of the root.
What does the multiplicity of a root in a polynomial indicate?
The degree of the polynomial
The sum of the coefficients
The sign of the quadratic coefficient
The number of times the root occurs
Multiplicity refers to how many times a given root appears in the factorization of a polynomial. In a repeated root scenario, the root appears more than once.
Given the quadratic equation 3x² - 6x + 3 = 0, determine the repeated root.
-1
1
2
0
Dividing the equation by 3 yields x² - 2x + 1 = 0, which factors as (x - 1)² = 0. This shows that the repeated root is 1.
For a quadratic equation to have a repeated root, its graph touches the x-axis at exactly one point. Which point does it touch if the quadratic is written as a(x - r)² = 0?
r
-r
0
2r
The vertex of the parabola in vertex form a(x - r)² is at (r, 0) when set equal to zero, indicating a repeated root at x = r. Thus, the graph touches the x-axis at this point.
What is the vertex of the parabola given by the quadratic equation in vertex form a(x - r)² = 0?
(r, a)
(0, a)
(r, 0)
(0, r)
In the vertex form, the expression (x - r)² shows that the x-coordinate of the vertex is r, and since the equation equals zero, the y-coordinate is 0. Hence, the vertex is at (r, 0).
If a quadratic equation has a repeated root r, what is its factorized form?
a(x - 2r)(x + 0.5r)
a(x - r)(x + r)
a(x² - r²)
a(x - r)²
A repeated or double root means the factor (x - r) appears twice. Therefore, the correct factorization is a(x - r)².
Given the quadratic equation x² + 6x + 9 = 0, which factorization correctly reveals its repeated root?
(x - 6)² = 0
(x + 3)² = 0
(x + 6)² = 0
(x - 3)² = 0
Expanding (x + 3)² yields x² + 6x + 9, confirming that the repeated root is -3. This factorization reveals the structure of a double root.
If a quadratic equation with a repeated root r is multiplied by a nonzero constant, how does it affect the location of the repeated root?
It only changes the multiplicity of the root.
It shifts the root to a different value.
It reverses the sign of the root.
The repeated root remains at r.
Multiplying a quadratic equation by a nonzero constant does not change its roots. The repeated root remains unchanged at r regardless of the scaling factor.
How does the graph of a quadratic equation with a repeated root differ from one with two distinct real roots?
It is symmetric about the x-axis.
It has no vertex.
It touches the x-axis at one point rather than crossing.
It lies entirely above the x-axis.
A quadratic with a repeated root touches the x-axis at its vertex, whereas one with distinct real roots crosses the x-axis at two points. This single contact point distinguishes the graph of a quadratic with a repeated root.
Consider the quadratic equation 2x² - 4x + 2 = 0. Which method is most efficient in revealing its repeated root?
Completing the square
Applying the Rational Root Theorem
Using synthetic division
Graphing
Completing the square transforms the equation quickly into the form 2(x - 1)² = 0, clearly showing the repeated root. This method is the most direct approach for equations with potential perfect square trinomials.
How is the derivative of a quadratic equation related to its repeated root when one exists?
The derivative is never zero at the repeated root.
The repeated root is also a zero of the derivative.
The derivative is independent of the repeated root.
The derivative determines the multiplicity of the repeated root.
For a quadratic with repeated roots, differentiating yields a linear equation whose root matches the repeated root of the original equation. This is because the vertex, which is the repeated root, is also the critical point where the derivative equals zero.
Which of the following statements is always true for a quadratic equation with repeated roots?
It has a negative leading coefficient.
It has two distinct real solutions.
It has a unique solution with b² - 4ac = 0.
It has a complex conjugate pair of roots.
For a quadratic equation to have repeated roots, the discriminant must be zero, ensuring a unique solution (with multiplicity two). This is a defining property of quadratic equations with double roots.
Determine the value of k such that the quadratic equation x² + (k - 4)x + (4 - 2k) = 0 has a repeated root.
4
0
2
-4
A quadratic equation has a repeated root when its discriminant is zero. For this equation, applying the discriminant formula leads to k² = 0, hence k must be 0.
The quadratic equation ax² + bx + c = 0 has a repeated root when b² - 4ac = 0. If a = 2 and b = -8, what value of c satisfies this condition?
8
-8
4
16
Substituting a = 2 and b = -8 into the discriminant formula gives 64 - 8c = 0, which solves to c = 8. This ensures the discriminant is zero and the quadratic has a repeated root.
In the parabola described by y = a(x - r)², if a is negative, what can be inferred about the repeated root?
The repeated root shifts to a value greater than r.
The repeated root remains at x = r.
The repeated root shifts to a value less than r.
The repeated root disappears when a is negative.
The sign of a affects the direction in which the parabola opens but does not change the location of its vertex. Therefore, the repeated root, which is at the vertex x = r, remains unchanged.
For the quadratic function f(x) = (x - 3)², what is the sum of the roots when counted with their multiplicities?
3
0
9
6
Since the function factors as (x - 3)², the repeated root is 3, counted twice, giving a sum of 3 + 3 = 6. This is consistent with Viète's formula for quadratic equations.
Determine the value of m for which the equation (m + 1)x² - 2(m - 2)x + (m - 1) = 0 has a repeated root.
4/5
0
1
5/4
Calculating the discriminant for the given equation leads to 4[(m - 2)² - (m² - 1)] = 4(-4m + 5). Setting this equal to zero gives -4m + 5 = 0, so m must equal 5/4 for the quadratic to have a repeated root.
0
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Study Outcomes

  1. Understand the definition and properties of repeated roots in quadratic equations.
  2. Identify quadratic equations that feature polynomial roots with multiplicity 2.
  3. Apply algebraic methods to solve quadratic equations with repeated roots.
  4. Analyze the role of discriminants in determining the nature of quadratic roots.
  5. Evaluate solutions to confirm the presence of repeated roots in quadratic expressions.

Quiz: Polynomial Multiplicity 2 Root Cheat Sheet

  1. Repeated root - A repeated root occurs when a quadratic equation yields a single solution counted twice, so the parabola barely kisses the x-axis at one point. You can spot it algebraically or graphically as the vertex sits right on the axis. This concept is key before moving on to more complex polynomials. Explore repeated roots on MathBitsNotebook
  2. Discriminant - The discriminant \(b^2 - 4ac\) holds the secret code to your roots by telling you how many distinct answers to expect. When the discriminant equals zero, those two mirror-image roots collapse into one. Crunching this quick test is your fast pass to identifying repeated roots. Discover discriminant insights on MathBitsNotebook
  3. Zero discriminant in the quadratic formula - Plugging into \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}\) is like opening two doors; but if \(\sqrt{b^2 - 4ac}=0\), both doors lead to the same hallway. You end up with \(x = -\tfrac\), a clear sign of a repeated root. This shortcut lets you dodge extra algebra steps! See zero discriminant in action on MathBitsNotebook
  4. Graphical tangent - On the graph, a repeated root looks like the parabola tipping gently and touching the x-axis at its vertex without ever crossing. It's like watching a skateboarder just graze the rail before heading back up. Spotting this visual cue helps you confirm algebraic work in a slam-dunk judge of correctness. Visualize tangent behavior on MathStackExchange
  5. Perfect square factor - When you factor a quadratic with a repeated root, you'll get a perfect square trinomial: \((x - r)^2 = 0\). That double-binomial shows the same root \(r\) is stuck there twice, simplifying your solve to \(x = r\). Mastering this pattern speeds up your factor hunts in exams. Master perfect square factorization on MathBitsNotebook
  6. Vieta's formulas - Vieta's formulas tell us the sum of roots is \(-\tfrac\) and the product is \(\tfrac\). When you have a repeated root \(r\), the sum becomes \(2r\) and the product \(r^2\), creating a consistent check for your work. This neat pattern is a handy cross-check on tricky problems. Apply Vieta's formulas on MathBitsNotebook
  7. Higher‑degree polynomials - Repeated roots aren't just quadratic quirks; they show up in cubic and quartic polynomials too. They mark tangency points where graphs only touch the x-axis, so recognizing them can save you from sign errors. Mastering these double-zeroes gives you superpowers with more advanced equations. Explore repeated roots on MathStackExchange
  8. Multiplicity - In algebra lingo, a repeated root has multiplicity of 2, meaning it shows up twice in the factorization. Higher multiplicities mean even flatter bounces on the x-axis - triples are pancake-flat! Understanding multiplicity deepens your graph-reading radar. Understand root multiplicity on Classful
  9. Factoring example - If you see \((x - 3)^2 = 0\), bingo - you've got a repeated root at \(x = 3\). This perfect square trinomial signposts that 3 is your only answer, twice over. Scanning for that squared pattern makes factoring a breeze. Spot perfect square trinomials on MathBitsNotebook
  10. Practice drills - Drilling repeated root problems strengthens your pattern‑spotting instincts and builds exam confidence. Each extra problem lights up your brain's algebraic radar for detecting double roots quickly. Before you know it, these once-tricky cases become your favorite parabolas to graph! Practice repeated root exercises on MathBitsNotebook
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