First and second degree equations

First degree equations

An equation of the form $ax + b = 0$ has:

• exactly one solution, $x = -\frac{b}{a}$, when $a \ne 0$;
• no solution when $a = 0$ and $b \ne 0$ (it says $b = 0$, which is false);
• every real number as a solution when $a = 0$ and $b = 0$ (it says $0 = 0$).

The last two cases appear in questions with a parameter, so do not divide by a coefficient that might be zero.

With fractions, multiply both sides by the least common denominator first. Put brackets round every numerator before multiplying, because the minus sign in front of a fraction applies to its whole numerator: $-\frac{x - 1}{4}$ becomes $-(x - 1) = -x + 1$ after clearing.

Second degree equations

A quadratic $ax^2 + bx + c = 0$ with $a \ne 0$ can be solved three ways.

Factoring is fastest when it works. Find two numbers with product $ac$ and sum $b$; when $a = 1$, two numbers with product $c$ and sum $b$. Then use the zero-product rule: if $PQ = 0$, then $P = 0$ or $Q = 0$.

Special forms need no formula. If $b = 0$, isolate $x^2$: $x^2 = 16$ gives $x = \pm 4$. If $c = 0$, factor out $x$: $x^2 = 3x$ gives $x(x - 3) = 0$, so $x = 0$ or $x = 3$.

The quadratic formula always works:

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

The discriminant counts the solutions

$\Delta$	Real solutions
$\Delta > 0$	two distinct
$\Delta = 0$	one (a repeated root, $x = -\frac{b}{2a}$)
$\Delta < 0$	none

Questions often ask "for which values of $k$ does the equation have two real solutions?" Set up $\Delta$ in terms of $k$ and solve the inequality; you never need the roots themselves.

Sum and product of the roots

If $x_1$ and $x_2$ are the roots of $ax^2 + bx + c = 0$, then

$$x_1 + x_2 = -\frac{b}{a}, \qquad x_1 x_2 = \frac{c}{a}.$$

These follow from writing $a(x - x_1)(x - x_2)$ and comparing coefficients. They answer questions about the roots without the formula, and they check a solution: for $2x^2 - 7x + 3 = 0$ the roots should add to $\frac{7}{2}$ and multiply to $\frac{3}{2}$. They describe the real roots only when $\Delta \ge 0$, so check that first.

Equations that reduce to quadratics

A biquadratic such as $x^4 - 13x^2 + 36 = 0$ becomes a quadratic with $t = x^2$: $t^2 - 13t + 36 = (t - 4)(t - 9) = 0$, so $x^2 = 4$ or $x^2 = 9$, and $x = \pm 2$ or $x = \pm 3$. Remember that $t = x^2$ cannot be negative: a negative value of $t$ gives no real $x$.

The unknown in a denominator

Before you clear denominators, write down the domain: the values that make any denominator zero are excluded. After solving, throw out any solution in that excluded set. Multiplying by an expression that can be zero can create such false solutions, which is why the check is not optional.

Never divide by the unknown

Dividing both sides of $x^2 = 3x$ by $x$ gives $x = 3$ and silently loses $x = 0$. Move everything to one side and factor instead.