4.8 The “generic” quadratic equation

Put all the $x$ terms on one side, and the number on the other.

Divide both sides by the coefficient of $x^2$ .

Add the same number to both sides. What number? Half the coefficient of $x$ , squared .

• What is the coefficient of $x$ ?
• What is ½ of that?
• What is that squared?

OK, now add that to both sides of the equation.

>This brings us to a “rational expressions moment”—on the right side of the equation you will be adding two fractions. Go ahead and add them!

Rewrite the left side as a perfect square.

Square root—but with a “plus or minus”! (*Remember, if $x^2=\text{25}$ , $x$ may be 5 or –5!)

Finally, add or subtract the number next to the $x$ .

${\mathrm{4x}}^2+\mathrm{5x}+1=0$

${\mathrm{9x}}^2+\text{12}x+4=0$

${\mathrm{2x}}^2+\mathrm{2x}+1=0$

In general, a quadratic equation may have two real roots , or it may have one real root , or it may have no real roots . Based on the quadratic formula, and your experience with the previous three problems, how can you look at a quadratic equation ${\text{ax}}^2+\text{bx}+c=0$ and tell what kind of roots it will have?