<p>Hi everyone! I'm having a problem with one question from the Math section, can anyone help, please?</p>
<p>(x-8)(x-k)=x^2-5kx+m</p>
<p>In the equation above, k and m are constants. If the equation is true for all values of x, what is the value of m?</p>
<p>(A) 8
(B) 16
(C) 24
(D) 32
(E) 40</p>
<p>-5k=-8-k</p>
<p>m=(-8)(-k)
-k= m/-8
k=m/8</p>
<p>-5(m/8)= -8-(m/8)
(-5m)/8= (-64-m)/8
-5m= -64-m
-4m= -64
m = 16</p>
<p>If anything is confusing, ask me.</p>
<p>how did u arrive at the first step of ur solution jeff?</p>
<p>First part:
Foil : (x-8)(x-k)=x²**-kx-8x**+8k=x²**-5kx**+m</p>
<p>-kx-8x = -5kx
(x)(-k-8)= (x)(-5k) ---------- x cancels out
-k-8 = -5k</p>
<p>Second part:
Foil : (x**-8**)(x**-k**)=x²-kx-8x**+8k**=x²-5kx**+m**
8k = m
OR
(-8)(-k) = m</p>
<p>B) 16</p>
<p>Easy</p>
<p>I guess I found an easier solution.
Since the equation is true for all values of x, let x=0
(x-8)(x-k)=x²-5kx+m
8k=m
k=m/8
Then, let x=8
0= 64-40k+m
0= 64-40(m/8)+m
0= 64-5m+m
0= 64-4m
-64 = -4m
m = 16</p>
<p>A more direct solution is available if you know the expressions for the sum of the roots and product of the roots for quadratic equations.</p>
<p>See [Nature</a> of Roots](<a href=“http://www.regentsprep.org/Regents/math/algtrig/ATE4/natureofroots.htm]Nature”>http://www.regentsprep.org/Regents/math/algtrig/ATE4/natureofroots.htm) or google “sum and product of roots of quadratic equation.”</p>
<p>The right side quadratic must equal the left side quadratic; both must have identical roots.</p>
<p>From the right side, the sum of the roots is 5k and the product of the roots is m.</p>
<p>From the left side, the sum of the roots is 8 + k and the product of the roots is 8k.</p>
<p>So 8k = m (product of roots) and 5k = 8 + k (sum of roots)…</p>
<p>4k = 8</p>
<p>16 = m</p>