Question

5. The coordinates of the vertex of the curve y = 1 - 2x - 4x² are:

6. If 2x² - 12x + 11 ≡ 2(x + b)² + c, then

                A. b = 6, c = -7               B. b = 3, c = -7          C. b = -3, c = -7               D. b = 3, c = 7

Answer 1

Ans 5:-

Given curve is-

$y=1-2x-4x^2$

$y=\dfrac{-1}{4}(x^2+\dfrac{x}{2}-\dfrac{1}{4})$

$y=-\dfrac{1}{4}(x^2+\dfrac{x}{2}+\dfrac{1}{16}-\dfrac{1}{16}-\dfrac{1}{4})$

$y=-\dfrac{1}{4}(x+\dfrac{1}{4})^2+\dfrac{5}{64}$

On Comparing the above equation with $y=a(x-h)^2+k$

We get $h=\dfrac{-1}{4},k=\dfrac{5}{64}$

We get the vertex as $(-\dfrac{1}{4},\dfrac{5}{64})$

Ans 6:-

Given, $2x^2 - 12x + 11 = 2(x + b)^2 + c$

$2x^2-12x+11=2(x^2+b^2+2xb)+c\\2x^2-12x+11=2x^2+2b^2+4xb+c$

On comparing coefficient of Both the sides

we get $4b=-12,2b^2+c=11$

$b=\dfrac{-12}{4}=-3$

$\Rightarrow 2b^2+c=11\\\Rightarrow c=11-2(-3)^2\\\Rightarrow c=11-2(9)\\\Rightarrow c=11-18=-7$

Hence The value of $b=-3 \text{ and } c=-7.$