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كويز تفاعلي: Quadratic Formula & The Discriminant Worksheet - Reveal
This worksheet focuses on the quadratic formula and the discriminant. It includes questions on identifying the discriminant formula, determining the number and type of roots based on the discriminant value, and solving various quadratic equations. Students will practice working with real and imaginary roots, as well as understanding the graphical significance of setting quadratic equations to zero to find x-intercepts.
يرجى الانتباه إلى أن المعلم قام بإعداد الأسئلة فقط، ولم يقم بإعداد الإجابات أو الشروحات المرفقة. وقد تم توليد الإجابات باستخدام تقنيات الذكاء الاصطناعي، لذلك قد تتضمن بعض الأخطاء أو عدم الدقة.
The discriminant is
A
ax2 + bx + c
B
b - 4ac
C
b2 - 4ac
D
b2 + 4ac
Explanation
The discriminant of a quadratic equation in the form ax2 + bx + c = 0 is defined as b2 - 4ac .
Determine the value of the discriminant and describe the number and type roots for the following: x2 + 7x + 13
A
101; 2 real roots
B
3; 2 real roots
C
-101; 2 imaginary roots
D
-3; 2 imaginary roots
Explanation
For the equation x2 + 7x + 13 , a=1, b=7, c=13 . The discriminant is b2 - 4ac = 72 - 4(1)(13) = 49 - 52 = -3 . Since the discriminant is negative, there are two imaginary roots.
Use the quadratic formula to solve 2x2 + 2x - 12 .
A
x = -2, 3
B
x = 2, 3
C
x = 2, -3
D
x = -2, -3
Explanation
Simplifying 2x2 + 2x - 12 = 0 by dividing by 2 gives x2 + x - 6 = 0 . Factoring gives (x+3)(x-2) = 0 , so x = 2 and x = -3 .
If the discriminant is negative, then the quadratic has:
A
1 Real Solution
B
2 Real Solutions
C
Half a Solution
D
2 Complex (imaginary) Solutions
Explanation
A negative discriminant means the term under the square root in the quadratic formula is negative, resulting in two complex (imaginary) solutions.
The quadratic equation can be used to solve quadratic equations that can or cannot be factored.
Explanation
The quadratic formula is a universal method that can solve any quadratic equation, whether it is factorable or not.
Solve the equation
2p2 - 3p - 3 = 0 using the Quadratic Formula.
A
\(\frac{3 \pm \sqrt{33}}{4}\)
B
\(\frac{-1 \pm \sqrt{13}}{2}\)
C
{2, 1}
D
\(\frac{3 \pm \sqrt{17}}{2}\)
Explanation
Using the formula \(\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) with a=2, b=-3, c=-3 : \(\frac{3 \pm \sqrt{(-3)^2 - 4(2)(-3)}}{2(2)} = \frac{3 \pm \sqrt{9 + 24}}{4} = \frac{3 \pm \sqrt{33}}{4}\).
Solve using the quadratic formula: f(x) = 2x2 - 4x + 7
A
\(x = (2 \pm 2i\sqrt{10})/2\)
B
\(x = (4 \pm \sqrt{40})/4\)
C
\(x = (2 \pm i\sqrt{10})/2\)
D
\(x = (2 \pm 2i\sqrt{40})/4\)
Explanation
For 2x2 - 4x + 7 = 0 , roots are \(\frac{4 \pm \sqrt{16 - 56}}{4} = \frac{4 \pm \sqrt{-40}}{4} = \frac{4 \pm 2i\sqrt{10}}{4} = \frac{2 \pm i\sqrt{10}}{2}\).
If the discriminant is equal to 0, how many solutions are there?
A
No solutions
B
1 Solution
C
2 Solutions
D
Infinite solutions
Explanation
When the discriminant is zero, the quadratic equation has exactly one real solution (a repeated root).
If the discriminant is equal to 4, how many solutions are there?
A
No Solutions
B
1 Solution
C
2 Solutions
D
Infinite Solutions
Explanation
When the discriminant is positive, the quadratic equation has two distinct real solutions.
Solve the equation
x2 - x - 2 = 0 using the Quadratic Formula.
A
\(\frac{-1 \pm \sqrt{10}}{3}\)
B
{2, -1}
C
{2, -3}
D
{1, -2}
Explanation
Factoring x2 - x - 2 = 0 gives (x-2)(x+1) = 0 , resulting in roots x=2 and x=-1 .
Use the quadratic formula to determine the solutions to the equation 2x2 - 9x - 35 = 0 .
A
x = 7/2, x = -6
B
x = -5/2, x = 5
C
x = -3/7, x = 6
D
x = -5/2, x = 7
Explanation
Using the quadratic formula: \(x = \frac{9 \pm \sqrt{81 - 4(2)(-35)}}{4} = \frac{9 \pm \sqrt{361}}{4} = \frac{9 \pm 19}{4}\). Solutions are 28/4 = 7 and -10/4 = -5/2 .
Solve x2 - 5x + 10 = 0 .
A
\(x = (5 - i\sqrt{15})/2, (5 + i\sqrt{15})/2\)
B
\(x = (5 - \sqrt{15})/2, (5 + \sqrt{15})/2\)
C
\(x = (5 - i\sqrt{65})/2, (5 + i\sqrt{65})/2\)
D
\(x = (5 - \sqrt{65})/2, (5 + \sqrt{65})/2\)
Explanation
The discriminant is 25 - 40 = -15 . Roots are \(\frac{5 \pm \sqrt{-15}}{2} = \frac{5 \pm i\sqrt{15}}{2}\).
Solve for x : 3x2 - 6x + 6 = 0
A
\(x = 1 \pm i\)
B
x = 3, -6
C
x = 4.8, 2.7
D
Undefined
Explanation
Divide by 3 to get x2 - 2x + 2 = 0 . The discriminant is 4 - 8 = -4 . Roots are \(\frac{2 \pm \sqrt{-4}}{2} = \frac{2 \pm 2i}{2} = 1 \pm i\).
Solve 2x2 - 36 = x
A
\(x = \frac{-9}{2}\) and 4
B
\(x = \frac{9}{2}\) and -4
C
x = 4 and -4
D
No real solution
Explanation
Rewrite as 2x2 - x - 36 = 0 . Using formula: \(x = \frac{1 \pm \sqrt{1 - 4(2)(-36)}}{4} = \frac{1 \pm 17}{4}\). Solutions are 18/4 = 4.5 and -16/4 = -4 .
Why are quadratic equations set equal to zero?
A
Because zero is an easy number to work with.
B
Because we are trying to find the x-intercepts and that happens when y = 0 .
C
Because setting it equal to zero helps us find the y-intercepts.
D
None of the above
Explanation
Setting a quadratic function equal to zero allows us to solve for x values where the graph intersects the horizontal axis (x-intercepts).
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