Quadratic equations in one unknown ：練習問題

Find the nature of the roots of the quadratic equation $$3x^2 - 4x + 2 = 0$$.

If one root of the quadratic equation \(5x^2 - 12x + k = 0\) is twice the other, find the value of \(k\).

If the roots of the quadratic equation \(x^2 - (2m+1)x + m^2 - 1 = 0\) are real and one root is twice the other, find the possible value(s) of \(m\).

Let \(\alpha\) and \(\beta\) be the roots of the quadratic equation \(x^2 - 5x + k = 0\) . If the sum of the squares of the roots is 13, i.e., \(\alpha^2 + \beta^2 = 13\) , find the value of the constant \(k\) .

If one root of the quadratic equation \(2x^2 - 7x + k = 0\) is twice the other, find the value of \(k\).

Form a quadratic equation in \(x\) with integer coefficients whose roots are 5 and -2.

If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(x^2 - 6x + (k+3) = 0\), and it is given that \(\alpha^2 + \beta^2 = 20\), find the value of the constant \(k\).

Find the range of values of $$k$$ such that the straight line $$y = x + k$$ does not intersect the parabola $$y = x^2 - x + 2$$.

Let $$\alpha$$ and $$\beta$$ be the roots of the quadratic equation $$2x^2 - 6x + 5 = 0$$.

(a) Find the sum of the roots, $$\alpha + \beta$$.

(b) Find the product of the roots, $$\alpha \beta$$.

(c) Hence, find the value of $$\frac{1}{\alpha} + \frac{1}{\beta}$$.

Consider the quadratic equation $$3x^2 - 6x + m = 0$$.

(a) Find the value of $$m$$ if the equation has exactly one real root.

(b) For the value of $$m$$ found in part (a), find this real root.