Find a so that f(x) = ax2 + 6x + 3 has two real roots.

Answer 1

A quadratic equation has real roots when the discriminate, b²-4ac, > zero, If b²-4ac=0 then theoretically it also has 2 roots, but they repeat. they have the same value.

Here b=6 c=3 so b²-4ac = 36-4a(3) =36-12a set that equal zero

36-12a=0 solve for a

12a=36

a=36/12=3 That's the a value for two repeated real roots

for a<3 f(x) has 2 real roots and they don't repeat

for z>3 f(x) has no real roots. It has two imaginary or complex roots

3x²+6x+3=0 divide through by 3 to get

x²+2x+1=0 then factor

(x+1)²=0 that factors to (x+1)(x+1)=0 where x+1=0 for both factor and

x=-1 is the real root that repeats

For a<3, such as a=-3

you have -3x²+6x+3 here the discriminate is 36+36=72 the two real roots are (-b+ and - sqr72)2/a or( -6+ and - 2^1/2(6))/-6 or 1 plus and minus (2)^1/2 = approximately 1+1.4 and 1-1.4 = 2.4, -0.4

But a cannot = 0, as then the quadratic collapses to a straight line, with only one real root, x=-1/2

Any other number a < 3 will have two real roots.

Graphically the functions will all be parabolas

For a>3, the parabola never touches the x-axis

For a=3 the parabola touches the x-axis only once

For a=0, the parabola degenerates into a vertical line which touches the x-axis only once. the closer a is to 0, the more narrow the parabola

For a<3, the parabola crosses the x-axis twice, the x values at that intersection are the real roots

Answer 2

As long as the discriminant is greater than or equal to 0, the equation will have real roots.

or

sqrt( 36 -4a(3) ) = sqrt( 36 -12a ) >= 0

if

36-12a>=0 then -12a>=-36 or 12a<=36

So a <= 3

Answer 3

Answer 4

According to the question the equation has two roots

As we know condition for real roots are,

D=√(b²-4ac) > 0, -------------(1)

given quadratic equation is

f(x)= ax²+6x+3

by comparing with standard equation ax²+bx+c=0,

we get

a=a,

b=6,

c=3,

Now plugging values in eqn (1),

√ b²-4ac > 0

=> √(6²-4*a*3) > 0

squaring both side and solving we get,

36 > 12a

=>a < 3

there a value is a= (-∞,3)

Please let me know if you have any confusion