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Quizzes > High School Quizzes > Electives

3.09 Quadratics Practice Quiz

Sharpen Your Skills With a Quadratic Equation Quiz

Difficulty: Moderate
Grade: Grade 9
Study OutcomesCheat Sheet
Colorful paper art promoting Quadratic Equation Showdown, a dynamic high school math quiz

Solve the quadratic equation x^2 - 5x + 6 = 0.
x = -2 and x = -3
x = 1 and x = 6
x = -1 and x = -6
x = 2 and x = 3
The quadratic factors as (x - 2)(x - 3) = 0, giving the solutions x = 2 and x = 3. Factoring is a quick method for solving such equations.
Determine the discriminant of the equation x^2 - 4x + 4 = 0.
8
-4
4
0
Calculating the discriminant b^2 - 4ac for this equation gives 16 - 16 = 0. A discriminant of zero indicates one repeated real root.
If a quadratic equation has a discriminant of 25, what does this indicate about its solutions?
It has one real solution
It has two distinct real solutions
It has no real solutions
It has complex solutions
A positive discriminant, like 25, means the quadratic equation has two distinct real solutions. This follows from the properties of the quadratic formula.
Factor the quadratic equation 2x^2 + 5x + 2 = 0 and find its solutions.
x = 1 and x = 2
x = -1/2 and x = -2
x = 1/2 and x = 2
x = -1 and x = -2
By factoring, the equation can be rewritten as (2x + 1)(x + 2) = 0, yielding x = -1/2 and x = -2. Factoring confirms the correct roots for the quadratic.
Find the vertex of the parabola defined by y = x^2 - 6x + 8.
(-3, -1)
(-3, 1)
(3, -1)
(3, 1)
The vertex is found using the formula h = -b/(2a). Here, h = 6/2 = 3 and substituting back gives k = -1, so the vertex is (3, -1).
Solve the quadratic equation x^2 + 6x + 8 = 0 by completing the square.
x = -2 and x = -4
x = 2 and x = 4
x = -1 and x = -8
x = 1 and x = 8
Completing the square transforms the equation into (x + 3)^2 = 1, which leads to x = -2 and x = -4 when taking the square root. This method is useful when factoring is not straightforward.
Determine the number of real solutions for the equation 3x^2 - 4x + 5 = 0.
No real solutions
Two distinct real solutions
One real solution
Two equal real solutions
The discriminant for this equation is calculated as 16 - 60 = -44, which is negative. A negative discriminant indicates that the equation has no real solutions, only complex ones.
Solve 4x^2 - 12x + 9 = 0 using the quadratic formula.
x = -3/2
x = 3/2
x = 0
x = 3
Using the quadratic formula reveals that the discriminant is zero, yielding the single solution x = 3/2. This indicates that the quadratic is a perfect square.
Construct a quadratic equation with roots 2 and 5.
x^2 - 7x + 10 = 0
x^2 - 2x + 5 = 0
x^2 - 3x + 10 = 0
x^2 + 7x + 10 = 0
A quadratic with the given roots can be expressed as (x - 2)(x - 5) = 0, which expands to x^2 - 7x + 10 = 0. This method is standard for constructing quadratics from known solutions.
Which expression represents the standard form of a quadratic equation?
ax^2 + bx + c = 0
ax^2 x + bx
x^2 + c = bx
ax + bx^2 + c
The standard form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants and a is nonzero. This form is essential for analyzing the properties of quadratic functions.
Express the quadratic function f(x) = x^2 + 4x + 1 in vertex form.
(x + 2)^2 - 3
(x - 2)^2 + 3
(x - 2)^2 - 3
(x + 2)^2 + 3
Completing the square on the quadratic function yields (x + 2)^2 - 3, which is the vertex form. This format clearly shows the vertex and the direction in which the parabola opens.
Use the quadratic formula to solve 5x^2 + 3x - 2 = 0.
x = 2 and x = -0.5
x = 0.4 and x = -1
x = 1 and x = -0.4
x = 0.5 and x = -2
The quadratic formula gives x = (-3 ± √49) / 10, which simplifies to x = 0.4 and x = -1. This method is particularly useful when the quadratic does not factor easily.
Determine the axis of symmetry for the quadratic function f(x) = 2x^2 - 8x + 3.
x = -2
x = -3
x = 2
x = 3
The axis of symmetry is found using the formula x = -b/(2a); here it computes to x = 2. This vertical line splits the parabola into two mirror images.
Find the discriminant of the quadratic equation 2x^2 + 3x - 5 = 0.
49
-49
25
41
Calculating the discriminant b^2 - 4ac gives 9 - 4(2)(-5) = 9 + 40 = 49. A positive discriminant indicates the presence of two distinct real roots.
Solve the quadratic equation x^2 - 7x + 12 = 0.
x = 2 and x = 6
x = -3 and x = -4
x = 1 and x = 12
x = 3 and x = 4
The equation factors into (x - 3)(x - 4) = 0, which gives the solutions x = 3 and x = 4. Factoring is an efficient method when the quadratic has simple integer roots.
For which pair of values of k does the quadratic equation x^2 + kx + 9 = 0 have exactly one real solution?
k = 6 and k = 9
k = 6 and k = -6
k = 0 and k = 6
k = -6 and k = 9
A quadratic equation has exactly one real solution when its discriminant is zero. Setting k^2 - 36 = 0 leads to k = 6 or k = -6.
Find the vertex of the parabola described by f(x) = -3x^2 + 12x - 7.
(-2, 5)
(2, 5)
(-2, -5)
(2, -5)
The vertex is calculated using h = -b/(2a) which gives 2, and then computing f(2) results in 5. This confirms the vertex as (2, 5) and the axis of symmetry as x = 2.
If the quadratic function f(x) = x^2 + px + q has its vertex on the y-axis, what must be the value of p?
2
-1
0
1
The x-coordinate of the vertex is given by -p/(2a). For the vertex to lie on the y-axis, this value must be 0, implying that p = 0. This is a direct consequence of the symmetry of quadratic graphs.
A quadratic function f(x) = ax^2 + bx + c has a maximum value. What does this tell us about the coefficient a?
a > 0
a can be any non-zero value
a = 0
a < 0
A quadratic function attains a maximum value when the parabola opens downward, which requires that the coefficient a be negative. This determines the concavity and the overall shape of the quadratic graph.
Solve the quadratic equation 6x^2 - x - 2 = 0.
x = 1/2 and x = -2/3
x = 2/3 and x = -1/2
x = 2 and x = -1
x = -2/3 and x = -1/2
Applying the quadratic formula to 6x^2 - x - 2 = 0 yields x = (1 ± √49)/12, which simplifies to x = 2/3 and x = -1/2. This method ensures accurate calculation of the roots.
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Study Outcomes

  1. Analyze the structure of quadratic equations to identify coefficients and constant terms.
  2. Apply various methods such as factoring, completing the square, and the quadratic formula to solve equations.
  3. Evaluate the discriminant to determine the number and nature of the roots.
  4. Verify obtained solutions by substituting them back into the original equation.

Quadratic Equation Quiz - 3.09 Review Cheat Sheet

  1. Understand the Standard Form of a Quadratic Equation - Think of \(ax^2 + bx + c = 0\) as the blueprint for all quadratics. Spotting \(a\), \(b\), and \(c\) quickly helps you decide which solving tool to grab. OpenStax: Standard Form Guide
  2. Master the Quadratic Formula - When factoring feels like chasing ghosts, use \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}\) to rescue your solutions. It's a reliable one‑stop shop for any quadratic root hunt. Wikipedia: Quadratic Formula
  3. Learn to Complete the Square - Turn \(x^2 + bx + c\) into a perfect square trinomial and watch solving become magical. This makeover trick is your VIP pass to elegant solutions. Wikipedia: Completing the Square
  4. Apply the Square Root Property - If you've got \(x^2 = k\), just go for \(x = \pm\sqrt\). It's like cutting through the middleman when no linear term is in sight. OpenStax: Square Root Property
  5. Utilize the Zero-Product Property - Remember: if \(ab = 0\), then \(a = 0\) or \(b = 0\). Factoring your equation into binomials and setting each to zero is half the battle won. Symbolab: Zero-Product Guide
  6. Interpret the Discriminant - The secret code \(b^2 - 4ac\) tells you if your roots are real buddies or complex strangers. Positive means two real hits, zero means a neat double root, negative means you're venturing into complex territory. Wikipedia: Discriminant Explained
  7. Graph Parabolas Accurately - A quadratic's story is drawn as a parabola - up for a smile or down for a frown. Nail the vertex, axis of symmetry, and intercepts to sketch with confidence. OpenStax: Parabola Essentials
  8. Identify the Vertex - This is the crown jewel of your parabola, at \(x = -\frac\). It marks the highest high or lowest low, so knowing it feels like unlocking secret level rewards. OpenStax: Finding the Vertex
  9. Determine the Axis of Symmetry - Slice your parabola in half with the vertical line \(x = -\frac\). It's the mirror that keeps both sides in perfect harmony. LibreTexts: Axis of Symmetry
  10. Practice Factoring Quadratic Equations - Expressing quadratics as \((x - p)(x - q)\) is like performing algebraic yoga - flexible and neat. The more you drill, the faster you'll snap into solutions. Symbolab: Factoring Practice
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