Quadratic Equations — Exercise 4.2

Rewrite each equation as a product equal to zero, then set each factor equal to zero.

(i) x²−3x−10=0

\[x^2-5x+2x-10=0\]

\[(x-5)(x+2)=0\]

(ii) 2x²+x−6=0

\[2x^2+4x-3x-6=0\]

\[(2x-3)(x+2)=0\]

(iii) √2 x²+7x+5√2=0

\[\sqrt2x^2+2x+5x+5\sqrt2=0\]

\[(\sqrt2x+5)(x+\sqrt2)=0\]

(iv) 2x²−x+1/8=0

Multiply by 8 to remove the fraction:

\[16x^2-8x+1=(4x-1)^2=0\]

(v) 100x²−20x+1=0

\[(10x-1)^2=0\]

(i) John and Jivanti’s marbles

Let John start with x marbles; Jivanti starts with 45−x. After each loses five, the counts are x−5 and 40−x.

\[(x-5)(40-x)=124\]

\[-x^2+45x-200=124\;\Rightarrow\;x^2-45x+324=0\]

\[(x-36)(x-9)=0\]

(ii) Toys made in one day

Let n be the number of toys. Cost per toy is ₹(55−n), and total cost is ₹750.

\[n(55-n)=750\;\Rightarrow\;n^2-55n+750=0\]

\[(n-25)(n-30)=0\]

If one number is x, the other is 27−x. Their product gives:

\[x(27-x)=182\;\Rightarrow\;x^2-27x+182=0\]

\[(x-13)(x-14)=0\]

Let the smaller integer be n. The next is n+1.

\[n^2+(n+1)^2=365\]

\[2n^2+2n-364=0\;\Rightarrow\;n^2+n-182=0\]

\[(n+14)(n-13)=0\]

The root n=−14 is excluded because the integers must be positive.

Let the base be b cm, so the altitude is b−7 cm and b>7. Apply Pythagoras:

\[b^2+(b-7)^2=13^2\]

\[2b^2-14b-120=0\;\Rightarrow\;b^2-7b-60=0\]

\[(b-12)(b+5)=0\]

Reject b=−5 because a length cannot be negative. Thus b=12 and altitude=12−7=5.

Count×unit cost equals the total cost:

\[n(2n+3)=90\;\Rightarrow\;2n^2+3n-90=0\]

\[2n^2+15n-12n-90=0\]

\[(2n+15)(n-6)=0\]

The roots are n=6 and n=−15/2. A count must be a positive integer, so only n=6 is valid.