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zhenek [66]
3 years ago
8

For what value of k does the equation (2k+1)x^2+2x=10x-6 have two real and equal roots?

Mathematics
1 answer:
pickupchik [31]3 years ago
5 0

Answer:

\displaystyle k = \frac{5}{6}

Step-by-step explanation:

We are given the equation:

\displaystyle (2k+1)x^2 + 2x = 10x - 6

And we want to find the value of <em>k</em> such that the equation has two real and equivalent roots.

Since the equation is a quadartic, we can find its discriminant (symbolized by Δ). Recall that:

  • If Δ < 0, we have no real roots (two complex roots).
  • If Δ > 0, we have two real roots.
  • And if Δ = 0, we have one real root, or two equivalent ones.

First, rewrite our equation:

(2k+1)x^2 -8x + 6 =0

The discriminant is given by:

\displaystyle \Delta = b^2 -4ac

In this case, <em>b</em> = -8, <em>a</em> = (2<em>k</em> + 1), and <em>c</em> = 6.

Therefore, the discriminant is given by:

\displaystyle \Delta = (-8)^2 - 4(2k+1)(6)

For it to have two equal roots, the discriminant must be zero. Hence:

\displaystyle 0 = (-8)^2 - 4(2k+1)(6)

Solve for <em>k: </em>

<em />\displaystyle \begin{aligned} \displaystyle 0 &= (-8)^2 - 4(2k+1)(6) \\ 0 &= 64 - 48k - 24 \\ 0 &= 40 - 48k \\ -40 &= -48k \\ \\ k &= \frac{5}{6} \end{aligned}<em />

<em />

Hence, the value of <em>k</em> is 5/6.

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