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Quizzes > High School Quizzes > Mathematics

Discriminant Practice Quiz

Boost quadratic skills through interactive challenges

Difficulty: Moderate
Grade: Grade 8
Study OutcomesCheat Sheet
Colorful paper art promoting a high school algebra quiz on discriminants of quadratic equations.

What is the formula for the discriminant of a quadratic equation in the form ax^2 + bx + c = 0?
b^2 - 4ac
4ac - b^2
b - 4ac
b^2 + 4ac
The discriminant is defined as b^2 - 4ac and is used to determine the nature of the quadratic's roots. This formula helps determine if the solutions are real or complex.
For the quadratic equation 3x^2 + 6x + 3 = 0, what is the discriminant?
0
36
9
-9
Calculating the discriminant gives 6^2 - 4*3*3 = 36 - 36 = 0. A zero discriminant indicates the quadratic has one repeated real root.
What does a discriminant value of 0 indicate about the roots of a quadratic equation?
It indicates one real repeated root.
It indicates two distinct real roots.
It indicates two complex roots.
It indicates one real and one complex root.
When the discriminant is zero, the quadratic equation has exactly one real solution, meaning both roots are the same (a repeated or double root).
For the equation x^2 - 5x + 6 = 0, what is the value of the discriminant?
1
0
25
-1
The discriminant is calculated as (-5)^2 - 4*1*6 = 25 - 24, which equals 1. This positive value confirms that the equation has two distinct real roots.
For the quadratic equation 2x^2 + x + 3 = 0, what is the discriminant?
-23
23
11
0
Using the formula, the discriminant is computed as 1^2 - 4*2*3 = 1 - 24, which equals -23. A negative discriminant indicates that the equation has two complex solutions.
Find the value of k for which the quadratic equation x^2 - 4x + k = 0 has one real repeated solution.
4
2
6
-4
Setting the discriminant to zero gives (-4)^2 - 4*1*k = 16 - 4k = 0, which simplifies to k = 4. This condition ensures that the quadratic has one repeated real root.
Determine the discriminant of the quadratic equation 5x^2 - 2x + 1 = 0.
-16
16
20
10
The discriminant is calculated as (-2)^2 - 4*5*1 = 4 - 20, which equals -16. A negative discriminant indicates the presence of two complex conjugate roots.
If a quadratic equation has a discriminant of 49, how many distinct real solutions does it have?
Two distinct real solutions
One repeated real solution
No real solutions
One real and one complex solution
A discriminant of 49 is positive, which means the equation has two distinct real roots. A positive discriminant always indicates two different solutions.
Given the quadratic equation 3x^2 + bx + 5 = 0, for what range of values of b will the equation have complex solutions?
All values of b such that -√60 < b < √60
All values of b such that |b| > √60
b equals ±√60
Any value of b, because b does not affect the nature of the roots
For complex solutions, the discriminant b^2 - 4*3*5 must be negative, leading to b^2 < 60. This means b must be between -√60 and √60.
For the quadratic equation 4x^2 + 12x + c = 0, find the value of c that results in equal real roots.
9
6
12
3
Setting the discriminant 12^2 - 4*4*c to zero yields 144 - 16c = 0. Solving for c gives c = 9, which results in one real repeated solution.
What is the effect on the discriminant of a quadratic equation if the coefficient a is doubled?
It becomes b^2 - 8ac
It remains b^2 - 4ac
It doubles to 2(b^2 - 4ac)
It is halved to (b^2 - 4ac)/2
Doubling a changes the discriminant from b^2 - 4ac to b^2 - 8ac. This alteration can affect the nature of the roots if ac is positive.
Under what condition on the coefficients of a quadratic equation does it have two distinct real roots?
b^2 - 4ac > 0
b^2 - 4ac = 0
b^2 - 4ac < 0
a = 0
A quadratic equation has two distinct real roots when its discriminant is positive, meaning b^2 - 4ac > 0. This condition ensures that the square root in the quadratic formula is real and nonzero.
If the discriminant of a quadratic equation is a perfect square (with integer coefficients), what inference can be made about its roots?
The equation has two distinct rational roots
The equation always has irrational roots
The equation has a repeated root
The equation has complex roots
When the discriminant is a perfect square and the coefficients are integers, the square root in the quadratic formula is rational. This results in two distinct rational roots.
For the equation 2x^2 + mx + 3 = 0 to have no real roots, what must be true about m?
m must be between -2√6 and 2√6
m must be less than -2√6
m must be greater than 2√6
m must equal 0
The quadratic has no real roots if the discriminant m^2 - 24 is negative, meaning m^2 < 24. This inequality implies that m is between -√24 and √24, and since √24 simplifies to 2√6, the correct range is -2√6 < m < 2√6.
Determine the discriminant of the quadratic equation 7x^2 + 2x - 5 = 0.
144
4
140
-144
Calculating the discriminant gives 2^2 - 4*7*(-5) = 4 + 140 = 144, which is a positive perfect square. This indicates the equation has two distinct real roots.
Given the quadratic equation ax^2 + bx + c = 0, if the equation is multiplied by a non-zero constant k, how does the discriminant change?
It is multiplied by k^2
It is multiplied by k
It remains unchanged
It is divided by k^2
Multiplying the quadratic by k transforms the coefficients to ka, kb, and kc. The discriminant then becomes (kb)^2 - 4(ka)(kc), which simplifies to k^2(b^2 - 4ac).
Which expression correctly represents the quadratic formula highlighting the role of the discriminant?
x = (-b ± √(4ac - b^2))/(2a)
x = (-b ± √(b^2 - 4ac))/(a)
x = (-b ± √(b^2 - 4ac))/(2a)
x = (-b ± (b^2 - 4ac))/(2a)
The quadratic formula is expressed correctly as x = (-b ± √(b^2 - 4ac))/(2a). The term under the square root, the discriminant, determines the nature of the roots.
For the quadratic equation 2x^2 + (k - 3)x + k = 0 to have a single repeated solution, what must k be?
k = 7 ± 2√10
k = 7 + 2√10
k = 7 - 2√10
k = 7
Setting the discriminant to zero leads to (k - 3)^2 - 4*2*k = 0, which simplifies to k^2 - 14k + 9 = 0. Solving this results in k = 7 ± 2√10, the condition for a repeated root.
If a quadratic function f(x) = ax^2 + bx + c touches the x-axis at exactly one point, what can be said about its vertex?
The vertex lies on the x-axis
The vertex lies above the x-axis
The vertex lies below the x-axis
The vertex is at the origin
When a quadratic touches the x-axis, it means that the parabola is tangent to the axis. This occurs when the vertex lies exactly on the x-axis, corresponding to a discriminant of zero.
A quadratic equation has a discriminant D. After increasing the constant term c by 5, the discriminant decreases by 20. What is the value of a?
1
2
5
4
Increasing c by 5 subtracts 4a*5 = 20a from the original discriminant. Since the decrease is 20, it implies that 20a = 20, and therefore a must equal 1.
0
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Study Outcomes

  1. Understand the structure of quadratic equations and identify their key components.
  2. Calculate the discriminant to examine the properties of quadratic equations.
  3. Analyze discriminant values to determine the nature of quadratic solutions.
  4. Interpret results to distinguish between real, repeated, and complex roots.
  5. Apply discriminant analysis to validate the number and type of solutions in quadratic scenarios.

Discriminant Practice Cheat Sheet

  1. Compute the Discriminant - Start by calculating Δ = b² − 4ac; it's your magic key to predicting root behavior. This simple step tells you whether solutions are real or complex without solving the whole equation. AnalyzeMath: Discriminants
  2. Two Real Roots (Δ > 0) - A positive Δ means your parabola cuts the x-axis twice, giving two distinct real solutions. For example, x² − 3x + 2 has Δ = 1, so you'll find two neat real roots. See worked examples
  3. One Repeated Root (Δ = 0) - When Δ equals zero, the quadratic just kisses the x-axis once, delivering a double root. In x² − 4x + 4, Δ = 0, so that single solution comes with extra style points. Learn more
  4. Complex Roots (Δ < 0) - A negative Δ signals no real intersections, but two imaginary buddies pop up instead. For x² + x + 1, Δ = −3, so get ready to work with i and its imaginary charm. Dive into complex solutions
  5. Root Nature Revealed - Beyond counting roots, Δ tells you if they're real or imaginary - crucial for understanding your quadratic's graph. It's the short path to confidence before exams. MathWarehouse: Discriminant Guide
  6. Rational vs. Irrational Roots - A positive Δ that's a perfect square (like 16) points to neat rational roots, while a non‑square (like 12) means irrational surprises. Great way to predict decimal craziness! See rational vs. irrational
  7. Quadratic Formula Power-Up - Δ sits under the square root in x = (−b ± √Δ)/(2a), controlling the ± magic. Master Δ and the formula practically solves itself. Review the formula
  8. Graphing Insight - Use Δ to know if your parabola hits the x‑axis two times, one time, or not at all - no drawing needed at first. It's graphing with superpowers. Graphing tips
  9. Practice Makes Perfect - Tackle varied quadratics and compute Δ to sharpen your skills. Try 2x² − 4x + 2 = 0 and predict its roots before solving. Practice problems
  10. Quick-Algebra Efficiency - Remember, Δ is your fast track to root info - no full solution required to know what you'll get. Impress classmates (and professors) with your speedy insights! Unlock more tricks
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