<span><span> x2-16x+48=0</span> </span>Two solutions were found :<span> x = 12 x = 4</span>
Step by step solution :<span>Step 1 :</span>Trying to factor by splitting the middle term
<span> 1.1 </span> Factoring <span> x2-16x+48</span>
The first term is, <span> <span>x2</span> </span> its coefficient is <span> 1 </span>.
The middle term is, <span> -16x </span> its coefficient is <span> -16 </span>.
The last term, "the constant", is <span> +48 </span>
Step-1 : Multiply the coefficient of the first term by the constant <span> <span> 1</span> • 48 = 48</span>
Step-2 : Find two factors of 48 whose sum equals the coefficient of the middle term, which is <span> -16 </span>.
<span><span> -48 + -1 = -49</span><span> -24 + -2 = -26</span><span> -16 + -3 = -19</span><span> -12 + -4 = -16 That's it</span></span>
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -12 and -4
<span>x2 - 12x</span> - 4x - 48
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (x-12)
Add up the last 2 terms, pulling out common factors :
4 • (x-12)
Step-5 : Add up the four terms of step 4 :
(x-4) • (x-12)
Which is the desired factorization
<span>Equation at the end of step 1 :</span> (x - 4) • (x - 12) = 0
<span>Step 2 :</span>Theory - Roots of a product :
<span> 2.1 </span> A product of several terms equals zero.<span>
</span>When a product of two or more terms equals zero, then at least one of the terms must be zero.<span>
</span>We shall now solve each term = 0 separately<span>
</span>In other words, we are going to solve as many equations as there are terms in the product<span>
</span>Any solution of term = 0 solves product = 0 as well.
Solving a Single Variable Equation :
<span> 2.2 </span> Solve : x-4 = 0<span>
</span>Add 4 to both sides of the equation :<span>
</span> x = 4
Solving a Single Variable Equation :
<span> 2.3 </span> Solve : x-12 = 0<span>
</span>Add 12 to both sides of the equation :<span>
</span> x = 12
Supplement : Solving Quadratic Equation Directly<span>Solving <span> x2-16x+48</span> = 0 directly </span>
Earlier we factored this polynomial by splitting the middle term. let us now solve the equation by Completing The Square and by using the Quadratic Formula
Parabola, Finding the Vertex :
<span> 3.1 </span> Find the Vertex of <span>y = x2-16x+48</span>
