Answer to Question #122899 in Algebra for khomotso

Question

( a) solve for k in the equation -8+44=-20k+80 and explain each step
( b) factorise the following quadratic expressions:
( i) x2 + 14x +24
( ii) t2+12t+35
( iii) m2 + 10m -24
( iv) p2 +8p -84
( v) y2 -4y-32
( vi) x2-7x-18
( vii) v2 - 9v +20
( viii) v2 -12v +32

( c) equate the expression in (b) above to 0 and find the two values of the variables for which the equation will be true

Answer 1

(a)

-8+44=-20k+80

or, 20k = 80 + 8 - 44 ( brought 20k term to L.H.S and all numerical terms to R.H.S )

or, 20k = 88 - 44 ( applied simplification rule to R.H.S)

or, 20k = 44

or, k = $\frac{44}{20}\\$ ( brought 20 to the denominator of R.H.S )

or, k = $\frac{11}{5}$ ( cancelling the common factor 4 from numerator and denominator).

Answer : $k = \frac{11}{5}.$

(b) (i)

$x^2+12x+24\\x^2+(12+2)x+24\\x^2+12x+2x+24\\x(x+12)+2(x+12)\\(x+12)(x+2)\space Answer$

(b)(ii)

$t^2+12t+35\\t^2+(7+5)t+35\\t^2+7t+5t+35\\t(t+7)+5(t+7)\\(t+7)(t+5)\space Answer$

(b)(iii)

$m^2+10m-24\\m^2+(12-2)m-24\\m^2+12m-2m-24\\m(m+12)-2(m+12)\\(m+12)(m-2)\space Answer$

(b)(iv)

$p^2+8p-84\\p^2+(14-6)p-84\\p^2+14p-6p-84\\p(p+14)-6(p+14)\\(p+14)(p-6)\space Answer$

(b)(v)

$y^2-4y-32\\u^2-(8-4)y-32\\y^2-8y+4y-32\\y(y-8)+4(y-8)\\(y-8)(y+4)\space Answer$

(b)(vi)

$x^2-7x-18\\x^2-(9-2)x-18\\x^2-9x+2x-18\\x(x-9)+2(x-9)\\(x-9)(x+2)\space Answer$

(b)(vii)

$v^2-9v+20\\v^2-(5+4)v+20\\v^2-5v-4v+20\\v(v-5)-4(v-5)\\(v-5)(v-4)\space Answer$

(b)(viii)

$v^2-12v+32\\v^2-(8+4)v+32\\v^2-8v-4v+32\\v(v-8)-4(v-8)\\(v-8)(v-4)\space Answer$

(c)

$(i) (x+12)(x+2)=0\\\implies either\space (x+12)=0\space or\space(x+2)=0\\\implies x=-12\space or\space x=-2\space$

$(ii) (t+7)(t+5)=0\\\implies either\space (t+7)=0\space or\space(t+5)=0\\\implies t=-7\space or\space t=-5\space$

$(iii) (m+12)(m-2)=0\\\implies either\space (m+12)=0\space or\space(m-2)=0\\\implies m=-12\space or\space m=2\space$

$(iv) (p+14)(p-6)=0\\\implies either\space (p+14)=0\space or\space(p-6)=0\\\implies p=-14\space or\space p=6\space$

$(v) (y-8)(y+4)=0\\\implies either\space (y-8)=0\space or\space(y+4)=0\\\implies y=8\space or\space y=-4\space$

$(vi) (x-9)(x+2)=0\\\implies either\space (x-9)=0\space or\space(x+2)=0\\\implies x=9\space or\space x=-2\space$

$(vii) (v-5)(v-4)=0\\\implies either\space (v-5)=0\space or\space(v-4)=0\\\implies v=5\space or\space v=4\space$

$(viii) (v-8)(v-4)=0\\\implies either\space (v-8)=0\space or\space(v-4)=0\\\implies v=8\space or\space v=4\space$