QuestionFactor the quadratic equation $3p^{2}+32p+20$ .

Studdy Solution

STEP 1

Assumptions1. The given equation is a quadratic equation in the form $ax^{} + bx + c$ . We need to factorize the equation into the form $(dx + e)(fx + g)$

STEP 2

The quadratic equation given is $p^{2}+32p+20$ .

STEP 3

To factorize the quadratic equation, we need to find two numbers that add up to the coefficient of $p$ (which is32) and multiply to give the product of the coefficient of $p^{2}$ (which is3) and the constant term (which is20).
So we need to find two numbers that satisfy the following conditions1. Their sum is equal to32.
2. Their product is equal to $3 \\times20 =60$ .

STEP 4

The two numbers that satisfy these conditions are20 and12, because1. $20 +12 =32$
2. $20 \\times12 =240$

STEP 5

Now, we rewrite the middle term of the quadratic equation (the term with $p$ ) as the sum of the terms $20p$ and $12p$ .
So the equation $3p^{2}+32p+20$ becomes $3p^{2}+20p+12p+20$ .

STEP 6

Next, we group the terms to factor by grouping $3p^{2}+20p+12p+20 = (3p^{2}+20p) + (12p+20)$

STEP 7

Now, we factor out the greatest common factor from each group $(3p^{2}+20p) + (12p+20) = p(3p+20) +4(3p+20)$

STEP 8

Notice that the terms in the parentheses are the same. We can factor out the common binomial term $(3p+20)$ $p(3p+20) +4(3p+20) = (p+4)(3p+20)$ So, the factored form of the quadratic equation $3p^{2}+32p+20$ is $(p+4)(3p+20)$ .

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