Quadratic equations: factoring, the formula and the discriminant

What makes an equation quadratic

A quadratic equation can be written as $ax^2 + bx + c = 0$ with $a \neq 0$. It can have two real solutions, one, or none. That is the key difference from a linear equation: there is usually more than one value of $x$ that works, and many questions turn on the second one.

Step one: make one side zero

Every method starts from $ax^2 + bx + c = 0$. Move all terms to one side first. The equation $x^2 = 5x + 14$ becomes $x^2 - 5x - 14 = 0$. Do not try to factor while terms are on both sides.

Factoring and the zero-product rule

If a product equals zero, at least one factor equals zero. So once a quadratic is factored, each bracket gives a root:

$$x^2 - 5x - 14 = (x - 7)(x + 2) = 0 \quad\Rightarrow\quad x = 7 \text{ or } x = -2$$

Watch the signs: the factor $(x + 2)$ gives the root $-2$, not $2$. Then substitute each root back into the original equation; both should work.

The rule needs zero on one side. $(x - 1)(x - 4) = 10$ does not mean $x - 1 = 10$ or $x - 4 = 10$. Expand, move the 10 across and factor again: $x^2 - 5x - 6 = 0$, so $(x - 6)(x + 1) = 0$ and $x = 6$ or $x = -1$.

Never divide by the variable

Faced with $3x^2 = 12x$, it is tempting to divide both sides by $x$ and write $3x = 12$. That loses a root. Dividing by $x$ is only allowed if $x \neq 0$, and here $x = 0$ is a solution. Instead, move everything to one side and factor: $3x^2 - 12x = 3x(x - 4) = 0$, so $x = 0$ or $x = 4$.

Square roots give two signs

If a square equals a positive number, the thing being squared can be positive or negative. From $(x - 3)^2 = 25$, $x - 3 = 5$ or $x - 3 = -5$, so $x = 8$ or $x = -2$. Writing only $x - 3 = 5$ is one of the most common ways to lose a root.

The quadratic formula

When the numbers do not factor neatly, use

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Write down $a$, $b$ and $c$ with their signs before substituting, and put negative values in brackets. The whole numerator, $-b$ and the root, is divided by $2a$.

The discriminant counts the roots

The expression under the root, $b^2 - 4ac$, is the discriminant.

$b^2 - 4ac$	Real solutions
positive	two different ones
zero	exactly one (a repeated root)
negative	none

You can answer "how many solutions" questions without solving at all. For $x^2 + 4x + 7 = 0$, the discriminant is $16 - 28 = -12$, so there are no real solutions.

Sum and product of the roots

For $ax^2 + bx + c = 0$ with roots $r$ and $s$:

$$r + s = -\frac{b}{a} \qquad rs = \frac{c}{a}$$

This follows from writing the quadratic as $a(x - r)(x - s)$ and expanding. It answers "what is the sum of the solutions" in one line, and it is a quick check on roots you found by factoring: for $x^2 - 5x - 14 = 0$ the roots $7$ and $-2$ add to $5$ and multiply to $-14$, as they should. Check that the discriminant is not negative before you rely on it for real roots.

Two roots in a comparison

When a quantitative comparison asks about a variable defined by a quadratic, find both roots and compare each with the other quantity. If one root makes Quantity A greater and the other makes Quantity B greater, the relationship cannot be determined. If the stem adds a condition such as $x > 0$, use it to discard a root before comparing.