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Vedmedyk [2.9K]
2 years ago
11

flvs hope If $x^2+bx+16$ has at least one real root, find all possible values of $b$. Express your answer in interval notation.

Enter your answer Focus Quadratic Inequalities 18 Week 5 Quests © 2018 Art of Problem Solving Terms Privacy Contact Us About Us ap
Mathematics
1 answer:
Galina-37 [17]2 years ago
5 0
If x^2+bx+16 has at least one real root, then the equation x^2+bx+16=0 has at least one solution. The discriminant of a quadratic equation is b^2-4ac and it determines the nature of the roots. If the discriminant is zero, there is exactly one distinct real root. If the discriminant is positive, there are exactly two roots. The discriminant of <span>x^2+bx+16=0 is b^2-4(1)(16). The inequality here gives the values of b where the discriminant will be positive or zero:
                                           b^2-4(1)(16) ≥ 0
                                                  </span><span>b^2-64 ≥ 0
                                             (b+8)(b-8) </span><span>≥ 0
The answer is that all possible values of b are in the interval (-inf, -8]∪[8,inf) because those are the intervals where </span>(b+8)(b-8) is positive.
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Use vieta's theorem to solve the problems.
Mnenie [13.5K]

Using Vieta's Theorem, it is found that c = 72.

<h3>What is the Vieta Theorem?</h3>
  • Suppose we have a quadratic equation, in the following format:

y = ax^2 + bx + c

  • The roots are p and q.

The Theorem states that:

p + q = -\frac{b}{a}

pq = \frac{c}{a}

In this problem, the polynomial is:

x^2 - 17x + c

Hence the coefficients are a = 1, b = -17.

Since the difference of the solutions is 1, we have that:

p - q = 1

p = 1 + q

Then, from the first equation of the Theorem:

p + q = -\frac{b}{a}

1 + q + q = 17

2q = 16

q = 8

p = 1 + q = 9

Now, from the second equation:

pq = \frac{c}{a}

72 = c

c = 72

To learn more about Vieta's Theorem, you can take a look at brainly.com/question/23509978

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2 years ago
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Lena [83]

Answer:

<u>Option 2</u>

(2,8), because the graph of the two equations intersects at this point

Step-by-step explanation:

Just looking at the graph, you can tell that the point (2,8) is where the two lines intersect.

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AnnyKZ [126]

Answer:

y = -1/3x - 1

Step-by-step explanation:

Let's solve for y.

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Step 2: Divide both sides by 3.

y=-1/3x-1

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\frac{-1}{3} x-1

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750

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because if the has 650 students and it is the greatest number

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