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The quadratic equation x^2 - 2kx + (k - 1) = 0 has roots α and β such that α^2 + β^2 = 4. Without solving the equation, find the possible values of the real number k.

We know in a quadratic x^2 +bx + c = 0, -b/a = α + β and c/a = αβ.

Therefore, α + β = -(-2k) = 2k, and αβ = k - 1. (Both are divided by the coefficient in front of x which is 1 so can be ignored.<...

Answered by Ralph T. • Maths tutor

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Consider the arithmetic sequence 2, 5, 8, 11, ... a) Find U101 b) Find the value of n so that Un = 152

Firstly, as with any question, make sure to check your formula book in order to find any relevant equations. In this case, the one most relevant to us is U_n = U₁ + d(n-1).
From...

Answered by Kirsty W. • Maths tutor

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IB exam question: Let p(x)=2x^5+x^4–26x^3–13x^2+72x+36, x∈R. For the polynomial equation p (x) = 0 , state (i) the sum of the roots; (ii) the product of the roots.

p(x) = 2x⁵+ x⁴– 26x³– 13x²+ 72x + 36
i) obtain the sum by using vieta's fomula: for p(x) the sum is hence -b/a hence, sum = -1/2
ii) obtain the ...

Answered by Niccolo M. • Maths tutor

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What is the equation of the tangent drawn to the curve y = x^3 - 2x + 1 at x = 2?

The equation of a two-dimensional non-vertical line can easily be determined if its gradient 'm' and a point on the line (x0, y0) is known, using the formula m = (y - y0) / (x - x0). The gradient of the t...

Answered by Mario Z. • Maths tutor

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