QUADRATIC EQUATIONS
Algebra
Taking Square Root
A pure quadratic equation can be solved by taking the square root of both sides of the equation.
Before taking the square root, the equation must be arranged with the x2 term isolated on the left-
hand side of the equation and its coefficient reduced to 1. There are four steps in solving pure
quadratic equations by taking the square root.
Step 1.
Using the addition and subtraction axioms, isolate
the x2 term on the left-hand side of the equation.
Step 2.
Using the multiplication and division axioms,
eliminate the coefficient from the x2 term.
Step 3.
Take the square root of both sides of the equation.
Step 4.
Check the roots.
In taking the square root of both sides of the equation, there are two values that satisfy the
equation. For example, the square roots of x2 are +x and -x since (+x)(+x) = x2 and
(-x)(-x) = x2. The square roots of 25 are +5 and -5 since (+5)(+5) = 25 and (-5)(-5) = 25. The
two square roots are sometimes indicated by the symbol ±. Thus,
. Because of this
25
±5
property of square roots, the two roots of a pure quadratic equation are the same except for their
sign.
At this point, it should be mentioned that in some cases the result of solving pure quadratic
equations is the square root of a negative number. Square roots of negative numbers are called
imaginary numbers and will be discussed later in this section.
Example:
Solve the following quadratic equation by taking the square roots of both sides.
3x2 = 100 - x2
Solution:
Step 1.
Using the addition axiom, add x2 to both sides of the equation.
3x2 + x2
= 100 - x2 + x2
4x2
= 100
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