多项式方程

Introduction: From Linear to Quadratic

Discriminant and Root Formula

When $\Delta=0$, how many real roots does the quadratic equation $ax^2+bx+c=0$ have?

Vieta's Theorem: Roots and Coefficients

What is the product $P$ of the roots of $2x^2+3x-5=0$?

Deriving the Root Formula by Completing the Square

After completing the square we get $\left(x+\dfrac{b}{2a}\right)^2=\dfrac{\Delta}{4a^2}$. What condition is required to take the square root?

Factorization Using the Roots

When $3x^2+7x+2$ is factored in the form $a(x-\alpha)(x-\beta)$, what is $a$?

Substitution for Biquadratic Equations

What substitution reduces the biquadratic equation $ax^4+bx^2+c=0$ to a quadratic equation?

Worked Example: Discriminant and Double Root

Find the two roots of $x^2-5x+6=0$, separated by semicolons from smallest to largest.

Let $\alpha$ and $\beta$ be the roots of $2x^2-x-3=0$. Use Vieta's theorem to find $\alpha^2+\beta^2$. Enter a number (as a fraction such as p/q).

Solve the biquadratic equation $x^4-5x^2+4=0$. List all real roots, separated by semicolons from smallest to largest.

How does the sign of $\Delta$ determine the number of real roots of the quadratic equation $ax^2+bx+c=0$?

Explain in your own words: why do we say 'one double root' when $\Delta=0$ but 'two distinct real roots' when $\Delta>0$? When $\Delta<0$ the equation has no roots over the reals; how is this conclusion seen from the process of completing the square?

Comprehensive Check