Advanced Math / Nonlinear Equations One Variable

Quadratic Equations in One Variable

Quadratic equations may be solved by factoring, taking square roots, or using the discriminant to reason about the number of real solutions. SAT-style items often ask for one solution, a parameter, or the number of real solutions rather than both roots.

Least you need to know

If $(x-r)(x-s)=0$, then $x=r$ or $x=s$.
If $(x-h)^2=m$, then $x-h=\sqrt m$ or $x-h=-\sqrt m$.
For $ax^2+bx+c=0$, the discriminant is $b^2-4ac$.
A quadratic has exactly one real solution when its discriminant is zero.

Key notation

$(x-4)(x+6)=0$ — A factored quadratic equation with solutions 4 and -6
$b^2-4ac$ — The discriminant of $ax^2+bx+c=0$

Worked example

To solve $x^2-5x=24$, first rewrite as $x^2-5x-24=0$.
Factor: $(x-8)(x+3)=0$.
The solutions are $8$ and $-3$.

Common mistakes

Move all terms to one side before factoring.
A squared equation can have two solutions.
When a question asks for the positive, lesser, greater, or nonzero solution, answer only that requested value.

How to recognize it

The problem asks for a solution to a quadratic equation.
The equation is already factored or can be factored.
The problem asks how many real solutions an equation has.

下一节推荐课程

通过 Radical and Rational Equations 继续学习此主题。

Radical and Rational Equations