If 2 is a root of $$ 2 k^{2}+k t+600 $$ find we whe of $$ k $$ and hence the otber rovt Fer what valuzis) of $$ p $$ witl the equation $$ x^{2}-2 p x-4 p-21-0 $$ bave real roors

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UpStudy AI · Accepted Answer

Given that 2 is a root of the equation
$$
2k^2 + kt + 600 = 0,
$$
we substitute $$ k=2 $$ into the equation:

$$
2(2)^2 + 2t + 600 = 0.
$$

Step 1. Simplify the equation:
$$
8 + 2t + 600 = 0 \quad \Longrightarrow \quad 2t + 608 = 0.
$$

Step 2. Solve for $$ t $$:
$$
2t = -608 \quad \Longrightarrow \quad t = -304.
$$

With $$ t = -304 $$, the equation becomes:
$$
2k^2 - 304k + 600 = 0.
$$

We can divide the entire equation by 2:
$$
k^2 - 152k + 300 = 0.
$$

Since 2 is a known root, let the other root be $$ r $$. The sum of the roots of a quadratic $$ k^2 - 152k + 300 = 0 $$ is given by:
$$
2 + r = 152,
$$
which leads to:
$$
r = 152 - 2 = 150.
$$

---

Now, consider the second part of the problem. We need to find for what values of $$ p $$ the equation
$$
x^2 - 2px - 4p - 21 = 0
$$
has real roots.

Step 3. Identify the coefficients for the quadratic in $$ x $$:
$$
a = 1, \quad b = -2p, \quad c = -4p - 21.
$$

Step 4. Compute the discriminant $$ \Delta $$:
$$
\Delta = b^2 - 4ac = (-2p)^2 - 4(1)(-4p - 21).
$$
$$
\Delta = 4p^2 + 16p + 84.
$$

Step 5. Factor out the constant:
$$
\Delta = 4(p^2 + 4p + 21).
$$

Step 6. Analyze the quadratic $$ p^2 + 4p + 21 $$. Its discriminant is:
$$
\Delta_p = (4)^2 - 4(1)(21) = 16 - 84 = -68.
$$

Since $$\Delta_p < 0$$ and the coefficient of $$ p^2 $$ is positive, the quadratic $$ p^2 + 4p + 21 $$ is always positive for all real $$ p $$.

Thus,
$$
\Delta = 4(p^2+4p+21) > 0 \quad \text{for all} \quad p \in \mathbb{R}.
$$

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Final Answers:
- The value of $$ t $$ is $$-304$$ and the other root of the first quadratic is $$150$$.
- The quadratic equation in $$ x $$ has real roots for all real values of $$ p $$.
- The other root of the first equation is 150. - The quadratic equation in $$ x $$ has real roots for all real values of $$ p $$.

If 2 is a root of find we whe of and hence the otber rovt
Fer what valuzis) of witl the equation bave real roors

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If 2 is a root of find we whe of and hence the otber rovt Fer what valuzis) of witl the equation bave real roors