اختبار اختيار من متعدد تفاعلي: Discriminant and Quadratic Formula - Reveal
The Discriminant and the Quadratic Formula are fundamental tools in Algebra for solving quadratic equations and determining the nature of their solutions. The quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), allows for the calculation of roots for any equation in the form \(ax^2 + bx + c = 0\). The discriminant, \(D = b^2 - 4ac\), is a specific part of the formula that provides insight into the type of solutions: a positive discriminant indicates two distinct real solutions, a discriminant of zero means there is exactly one real solution, and a negative discriminant results in two imaginary (complex) solutions. Mastering these algebraic techniques is essential for analyzing quadratic functions and their graphs.
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اختبار شهادة تدريبي مؤقت للصف والمادة والفصل نفسه.
اختر إجابة واحدة لكل سؤال. عند الاختيار ستظهر النتيجة فورًا: الأخضر صحيح، والأحمر خطأ، وسيظهر تفسير الإجابة مباشرة إن كان متوفرًا. وبعد آخر سؤال ستظهر الدرجة النهائية تلقائيًا.
Solve using the Quadratic Formula. x2 + 5x - 104 = 0
Explanation
Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) with a=1, b=5, c=-104, we calculate \(x = \frac{-5 \pm \sqrt{25 - 4(1)(-104)}}{2} = \frac{-5 \pm 21}{2}\). This gives \(x = \frac{16}{2} = 8\) and \(x = \frac{-26}{2} = -13\).
Question 2
Points: 1
Determine the values of a, b, and c for the quadratic equation: 4x2 - 8x = 3.
Explanation
To identify a, b, and c, the equation must be in standard form ax2 + bx + c = 0. Rearranging 4x2 - 8x = 3 gives 4x2 - 8x - 3 = 0. Thus, a = 4, b = -8, and c = -3.
Question 3
Points: 1
If the discriminant is zero you will have
Explanation
When the discriminant b2 - 4ac = 0, the quadratic formula yields a single real value \(x = \frac{-b}{2a}\), meaning there is exactly one real solution.
Question 4
Points: 1
The discriminant is
Explanation
In algebra, the discriminant of a quadratic equation ax2 + bx + c = 0 is the expression found under the radical in the quadratic formula, defined as b2 - 4ac.
Question 5
Points: 1
If the discriminant is positive, then the solution will be
Explanation
A positive discriminant (b2 - 4ac > 0) means the quadratic equation has two distinct real number solutions because the radical part of the formula results in a real number being added to and subtracted from the numerator.
Question 6
Points: 1
If the discriminant is negative, then the solution will be
Explanation
A negative discriminant (b2 - 4ac < 0) involves taking the square root of a negative number, which leads to two complex (imaginary) solutions.
Question 7
Points: 1
What does the discriminant tell us?
Explanation
The value of the discriminant determines whether a quadratic equation has real or imaginary roots and how many roots exist (one or two).
Question 8
Points: 1
Use the quadratic formula to solve f(x) = 2x2 + 2x + 12.
Explanation
Although the equation provided in the document is 2x2 + 2x + 12, the answer key identifies \(\{2, -3\}\) as correct. These are the roots for 2x2 + 2x - 12 = 0, as 2(2)2 + 2(2) - 12 = 8 + 4 - 12 = 0 and 2(-3)2 + 2(-3) - 12 = 18 - 6 - 12 = 0. There is a likely sign typo in the source document equation.
Question 9
Points: 1
What are the solutions to x2 - 3x = 5?
Explanation
First, write the equation in standard form: x2 - 3x - 5 = 0. Using the quadratic formula with a=1, b=-3, c=-5: \(x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-5)}}{2(1)} = \frac{3 \pm \sqrt{9 + 20}}{2} = \frac{3 \pm \sqrt{29}}{2}\).
Question 10
Points: 1
Use the quadratic formula to find the solutions for y = -x2 - 5x + 12
Explanation
Set y = 0 and identify coefficients: a = -1, b = -5, c = 12. Applying the formula: \(x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(-1)(12)}}{2(-1)} = \frac{5 \pm \sqrt{25 + 48}}{-2} = \frac{5 \pm \sqrt{73}}{-2}\).
Question 11
Points: 1
Use the quadratic formula to find the solutions for: 6x2 + 5x - 4 = 0
Explanation
Using a=6, b=5, c=-4: \(x = \frac{-5 \pm \sqrt{25 - 4(6)(-4)}}{2(6)} = \frac{-5 \pm \sqrt{25 + 96}}{12} = \frac{-5 \pm 11}{12}\). The solutions are \(x = \frac{6}{12} = 1/2\) and \(x = \frac{-16}{12} = -4/3\).
Question 12
Points: 1
Use the quadratic formula to find the solutions for: 5x + 2 = 3x2
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