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.
درس پیشنهادی بعدی
این موضوع را با Radical and Rational Equations ادامه دهید.
Radical and Rational Equationsدرسهای مرتبط
با درسهای نزدیک در همان موضوع ادامه دهید.