Key Points

A quadratic equation in the variable $x$ is an equation of the form $ax^2 + bx + c = 0$, where $a, b, c$ are real numbers and it is critical that $a eq 0$. This is known as the standard form of the equation.

A quadratic equation in the variable $x$ has the standard form $ax^2 + bx + c = 0$, where $a, b, c$ are real numbers and the coefficient of the squared term, $a$, must not be zero ($a \neq 0$). The degree of this equation is 2.

A real number $\alpha$ is called a root or a solution of the quadratic equation $ax^2 + bx + c = 0$ if $a\alpha^2 + b\alpha + c = 0$. The roots of the equation are the same as the zeroes of the corresponding quadratic polynomial, and a quadratic equation can have at most two roots.

A real number $\alpha$ is called a root or a solution of the quadratic equation $ax^2 + bx + c = 0$ if $a\alpha^2 + b\alpha + c = 0$. The roots of the equation are the same as the zeroes of the polynomial $ax^2 + bx + c$.

To find the roots by factorisation, the quadratic expression $ax^2 + bx + c$ is split into two linear factors. If the equation can be written as $(px+q)(rx+s) = 0$, the roots are found by setting each factor to zero, giving $x = -\frac{q}{p}$ and $x = -\frac{s}{r}$.

A quadratic equation can have at most two roots. This corresponds to the degree of the quadratic polynomial, which is 2.

If the quadratic polynomial $ax^2 + bx + c$ can be factorized into a product of two linear factors, the roots can be found by equating each factor to zero. For example, if $(x-p)(x-q)=0$, the roots are $x=p$ and $x=q$.

For any quadratic equation $ax^2 + bx + c = 0$, the roots can be found using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. This formula is valid provided $b^2 - 4ac \geq 0$.

The expression $D = b^2 - 4ac$ is called the discriminant of the quadratic equation. The value of the discriminant determines the nature of the roots without actually solving the equation.

For any quadratic equation $ax^2 + bx + c = 0$, the roots are given by the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. This formula is applicable only when $b^2 - 4ac \geq 0$.

The expression $D = b^2 - 4ac$ is called the discriminant of the quadratic equation. It determines the nature of the roots without actually solving the equation.

A quadratic equation has two distinct real roots if the discriminant is positive. That is, if $b^2 - 4ac > 0$.

If the discriminant $b^2 - 4ac > 0$, the quadratic equation has two distinct real roots. These roots are unequal.

A quadratic equation has two equal real roots (also called coincident roots) if the discriminant is zero. That is, if $b^2 - 4ac = 0$.

If the discriminant $b^2 - 4ac = 0$, the quadratic equation has two equal real roots (also called coincident roots). Each root is equal to $-\frac{b}{2a}$.

When a quadratic equation has two equal roots (when $b^2 - 4ac = 0$), both roots are identical and are given by the formula $x = -\frac{b}{2a}$.

If the discriminant $b^2 - 4ac < 0$, the quadratic equation has no real roots. The roots are complex numbers, which are not typically covered at this level.

A quadratic equation has no real roots if the discriminant is negative. That is, if $b^2 - 4ac < 0$, because the square root of a negative number is not a real number.

To determine if an equation is quadratic, you must first simplify it and write it in the standard form $ax^2 + bx + c = 0$. An equation might appear cubic or linear initially but simplify to a quadratic form.

To verify if an equation is quadratic, first simplify it and write it in the standard form $ax^2 + bx + c = 0$. The equation is quadratic only if the highest power of the variable is 2 and the coefficient $a$ is non-zero.