Answer to Question #163838 in Algebra for Daarren

Question

11. The range of values of a for which the equation ax²+2x - 2 = 0 has real roots is:

                 A. a ≤ 1/2            B. a ≥ 1/2            C. a ≤ -1/2        D. a ≥ -1/2

 

12. if the graph of y =x² - 8x + m cuts the x-axis in two distinct points then

               A. m < !6                      B. m > 16                  C. m ≤ 16                  D. m ≥ 16

Answer 1

11.Ans:- For an equation $ax^2+bx+c=0$ , $b^2-4ac$ is called the discriminant and helps in determining the nature of the roots of a quadratic equation. If $b^2-4ac>0$ , the roots are real and distinct. If $b^2-4ac=0$ , the roots are real and equal.

Given that $ax^2+2x-2=0$ has real roots.

So, $b^2-4ac\ge0$ here $b=2 , c=-2$

$\Rightarrow$ ${2}^2-4\times{a}\times{-2}\ge0$

$\Rightarrow$ $a\ge$ $-\dfrac{1}{2}$

Therefore, option (D) is correct.

12.Ans:-

condition for the quadratic equation cuts two distinct points on $x$ -axis is $\dfrac{-D}{4a}\le0$ where D is $b^2-4ac$.

$\Rightarrow$ $-\dfrac{8^2-4\times{1}\times{m}}{4\times1}\le0$

$\Rightarrow$ $m\le16$

therefore option (C) is correct