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Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients:$8x^2-22x-21$.
Given: A quadratic polynomial: $8x^2-22x-21$.
To do: To find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
Solution:
Given polynomial is:
$8x^2-22x-21$
$=8x^2-28x+6x-22x$
$=( 8x^2-28x)+( 6x-22x)$
$=4x( 2x-7)+3( 2x-7)$
$=( 4x+3)( 2x-7)$
Now,
If $4x+3=0$
$\Rightarrow x=-\frac{3}{4}$
If $2x-7=0$
$\Rightarrow x=\frac{7}{2}$
On comparing $8x^2-22x-21$ with $ax^2+bx+c$
$a=8,\ b=-22,\ c=-21$
$\alpha=-\frac{3}{4}$, $\beta=\frac{7}{2}$
Sum of zeroes $( \alpha+\beta)=-\frac{b}{a}$
$\Rightarrow -\frac{3}{4}+\frac{7}{2}=\frac{22}{8}$
$\Rightarrow \frac{22}{8}=\frac{22}{8}$
Product of zeroes $( \alpha\times\beta)=\frac{c}{a}$
$-\frac{3}{4}\times\frac{7}{2}=\frac{-21}{8}$
$\frac{-21}{8}=\frac{-21}{8}$
Hence, Verified.
Updated on: 10-Oct-2022
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