A quadratic equation is a one-variable polynomial equation of the second degree whose general form is:
ax2 + bx + c = 0,
where a, b and c are constants with a ≠ 0, and x represents a variable. A quadratic equation has two solutions, called roots. The two solutions may be or may not be distinct and may or may not be real. A quadratic equation can be solved by factoring. The method of factoring explained below is also called as decomposition method.
Let’s factor the equation 3x2 + 8x + 4 using the steps given below.
Here, a = 3, b = 8 and c = 4
Step 1: Multiply the term a and the term c
3 × 4 = 12
Step 2: Find two numbers whose product is equal to the product of terms a and c (12) and add up to be term b (8)
6 × 2 = 12 and 6 + 2 = 8
Step 3: Rewrite the middle term with these two numbers
3x2 + 6x + 2x + 4
Step 4: Factor the first two terms and last two terms separately
3x2 + 6x factor into 3x(x + 2)
2x + 4 factor into 2(x + 2)
Organize the equation so that you can take out the greatest common factor of the first two terms and the last two terms.
3x2 + 6x + 2x + 4 = 3x(x + 2) + 2(x + 2)
Both factored groups should be the same. (x + 2)
Step 5: Add the GCF’s together and enclose them in parentheses next to the factored group.
(3x + 2)(x + 2)
So (3x + 2) and (x + 2) are the factors if the quadratic 3x2 + 8x + 4.
Factors can be solved by equating to 0 to find the roots of the equation.
3x + 2 = 0; x + 2 = 0
3x = -2; x = -2
x = -2/3; x = -2
-2/3 and -2 are the roots of the equation 3x2 + 8x + 4
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