Let <em>a</em> and <em>b</em> be the zeroes of <em>x</em>² + <em>kx</em> + 12 such that |<em>a</em> - <em>b</em>| = 1.
By the factor theorem, we can write the quadratic in terms of its zeroes as
<em>x</em>² + <em>kx</em> + 12 = (<em>x</em> - <em>a</em>) (<em>x</em> - <em>b</em>)
Expand the right side and equate the coefficients:
<em>x</em>² + <em>kx</em> + 12 = <em>x</em>² - (<em>a</em> + <em>b</em>) <em>x</em> + <em>ab</em>
Then
<em>a</em> + <em>b</em> = -<em>k</em>
<em>ab</em> = 12
The condition that |<em>a</em> - <em>b</em>| = 1 has two cases, so without loss of generality assume <em>a</em> > <em>b</em>, so that |<em>a</em> - <em>b</em>| = <em>a</em> - <em>b</em>.
Then if <em>a</em> - <em>b</em> = 1, we get <em>b</em> = <em>a</em> - 1. Substitute this into the equations above and solve for <em>k</em> :
<em>a</em> + (<em>a</em> - 1) = -<em>k</em> → 2<em>a</em> = 1 - <em>k</em> → <em>a</em> = (1 - <em>k</em>)/2
<em>a</em> (<em>a</em> - 1) = 12 → (1 - <em>k</em>)/2 • ((1 - <em>k</em>)/2 - 1) = 12
→ (1 - <em>k</em>)²/4 - (1 - <em>k</em>)/2 = 12
→ (1 - <em>k</em>)² - 2 (1 - <em>k</em>) = 48
→ (1 - 2<em>k</em> + <em>k</em>²) - 2 (1 - <em>k</em>) = 48
→ <em>k</em>² - 1 = 48
→ <em>k</em>² = 49
→ <em>k</em> = ± √(49) = ±7
The question didn't make sense, could you add more to it?
Assuming larger number = x,
smaller number = y,
x = 2y - 3
Larger number = 45, smaller number = 24
Hope this helps. - M
Answer is: <span>m<POS=60 m<POQ=120</span>
Answer:
hello : the factorize completely is :( 5x+2a)(x-2a)
Step-by-step explanation:
the first solution :
note : Ax² + Bx + C = A( x - x1)(x - x2) if : Δ ≥ 0 and : Δ = B² - 4AC
x1 = ( - B -√Δ)/2A x2 = ( - B +√Δ)/2A
in this execice : A = 5 B = -8a C= - 4a²
calculate Δ = ( -8a)² - 4 (5)(-4a²) = 64a² +80a² = 144a² = (12a)²
x1 = ( 8a - 12a ) / 10 = -4a/10
x1 = -2a/5
x2 = ( 8a +12a ) / 10 = 20a/10
x2 =2a
put x1 and x2 in expression A( x - x1)(x - x2)
you have : 5( x +2a/5)(x -2a)= (5x+2a)(x-2a)
verify : (5x+2a)(x-2a) = 5x² -10ax +2ax -4a² = 5x² -8ax - 4a²
the second solution :
5x² -8ax - 4a² = 5x² + (-10ax +2ax) -4a² because
-8ax =-10ax +2ax
5x² -8ax - 4a² =(5x² + -10ax) +(2ax -4a²)
5x (x - 2a) +2a (x -2a) = (x -2a)(5x+2a)