Solving quadratic equations

If $AB=0$, then $A=0$ or $B=0$. For $(x+2)(x-3)=0$, $x+2=0$ or $x-3=0$, so $x=-2,\ 3$. The sign of each solution is opposite to the sign inside its parentheses.

Conversely, an equation with solutions $x=\alpha,\ \beta$ is $(x-\alpha)(x-\beta)=x^2-(\alpha+\beta)x+\alpha\beta=0$. With solutions $-5,\ -2$, it is $x^2+7x+10=0$.

When factoring is hard, use the quadratic formula. $a$, $b$ and $c$ are the coefficient of $x^2$, the coefficient of $x$ and the constant term, read together with their signs.

Examples

1. 01What is a in the quadratic formula?$\displaystyle 3x^2-6x-1 = 0$AnswerHide

   Answer$\displaystyle 3$

   Steps

   1. $\displaystyle a$$\displaystyle = 3$$\displaystyle b$$\displaystyle = -6$$\displaystyle c$$\displaystyle = -1$
2. 02What are the solutions?$\displaystyle (x+2)(x-3) = 0$AnswerHide

   Answer$\displaystyle x=-2,\ 3$

   Steps

   1. $\displaystyle x+2 = 0$$\displaystyle \Rightarrow x = -2$$\displaystyle x-3 = 0$$\displaystyle \Rightarrow x = 3$
3. 03Which is the larger root?$\displaystyle (x+1)(x-4) = 0$AnswerHide

   Answer$\displaystyle 4$

   Steps

   1. $\displaystyle x+1 = 0$$\displaystyle \Rightarrow x = -1$$\displaystyle x-4 = 0$$\displaystyle \Rightarrow x = 4$
4. 04What is b in the quadratic formula?$\displaystyle 3x^2+x+4 = 0$AnswerHide

   Answer$\displaystyle 1$

   Steps

   1. $\displaystyle a$$\displaystyle = 3$$\displaystyle b$$\displaystyle = 1$$\displaystyle c$$\displaystyle = 4$
5. 05What are the solutions?$\displaystyle (x+4)(x-1) = 0$AnswerHide

   Answer$\displaystyle x=-4,\ 1$

   Steps

   1. $\displaystyle x+4 = 0$$\displaystyle \Rightarrow x = -4$$\displaystyle x-1 = 0$$\displaystyle \Rightarrow x = 1$
6. 06What are the solutions?$\displaystyle x^2 = 16$AnswerHide

   Answer$\displaystyle x=\pm 4$

   Steps

   1. $\displaystyle 4^2$$\displaystyle = 16$$\displaystyle (-4)^2$$\displaystyle = 16$
   2. Both square to the same value, so
   3. $\displaystyle x$$\displaystyle = \pm 4$

FAQ

Q1Are the solutions of $(x+4)(x-1)=0$ $x=4,\ -1$?AnswerHide

A

The signs are reversed. $x+4=0$ gives $x=-4$, and $x-1=0$ gives $x=1$, so the solutions are $x=-4,\ 1$.

Q2For $3x^2-6x-1=0$, is $b=6$ in the quadratic formula?AnswerHide

A

$b$ includes its sign, so $b=-6$. Substitute $a=3$, $b=-6$, $c=-1$.