If x_1 and x_2 are the two roots of the quadratic equation x^2-px+q=0, (x_1>x_2). Find the values of the following expressions.

Let m and n be the two roots of the equation x^2 − 15x + 28 = 0. Find the value of (m + 1)(n + 1)

Let x_1 and x_2 be the roots of the equation x^2 + 3x + 1 = 0. Compute

\Big( \dfrac{x_1}{x_2+1} \Big) ^2+\Big( \dfrac{x_2}{x_1+1} \Big) ^2

Let x_1 and x_2 be the roots of the equation x^2 − (a + d)x + (ad − bc) = 0. Show that x^3_1 and x^3_2 are the roots of the equation

y^2 − (a^3 + d^3 + 3abc + 3bcd)y + (ad − bc)^3 = 0

The depressed cubic equation x^3 +px + 28=0 has three distinct roots. Two of these roots sum to 2. Find the the value of p.

5 Equations to Show Solution Sets

Roots of Cubic Equations

If x+y = 1, x^4+y^4 = 881, find the value of xy

Solve the equation

x^2(1+x)^2+x^2=8(1+x)^2

Solve the cube root equation

\sqrt[3]{14+\sqrt{x} } +\sqrt[3]{14-\sqrt{x} } =4

Solve the equation

(x+2)^4+(x+1)^4=97

Solve the equation

(x+2)^4+x^4 = 82

Solve the equation

\dfrac{1}{x^2}+\dfrac{1}{(x+2)^2}=\dfrac{10}{9}