If the roots of ax
2 + bx + c = 0 are equal then c ?
Answers
Answer:
the equation ax^2+bx+c = 0, the roots are
x1 = [-b+(b^2–4ac)^0.5]/2a and
x2 = [-b-(b^2–4ac)^0.5]/2a
If the roots are to be equal, then x1 = x2, or
[-b+(b^2–4ac)^0.5]/2a = [-b-(b^2–4ac)^0.5]/2a, or
+(b^2–4ac)^0.5] =-(b^2–4ac)^0.5, or
2(b^2–4ac) = 0, or
b^2 = 4ac, or
b = 2(ac)^0.5.
So the equation will have equal roots when b = 2(ac)^0.5. And, the equation becomes
ax^2+2(ac)^0.5 x+c = 0 and the two equal roots will be
x3=[-2(ac)^0.5 +(4ac-4ac)^0.5]/2a or
x3=x4 = -(c/a)^0.5
The equal roots will be x = -(c/a)^0.5, when b = 2(ac)^0.5.
Answer: -b/2a
Solution:
The solution of the quadratic equation
ax² + bx + c = 0
is given by the formula
x = [-b ± √(b² - 4ac)]/2a
Let the roots be denoted by x1 and x2, so that
x1 = [-b + √(b² - 4ac)]/2a, x2 = [-b - √(b² - 4ac)]/2a
If b² - 4ac = 0, x1 and x2 are real and equal, each reducing to -b/2a.
Hence the equal roots are -b/2a.
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SHORT ANSWER AT THE BOTTOM IF YOU DON'T WANT THE FULL EXPLENATION.
Remember this bad boy of an equation?:
x = -b +/- (square root) b^2 - 4ac
. — — — — — — — — — — — —
. ……………….. 2a
The determinant is b^2 - 4ac
Because it determines how many answers.
•If b^2 - 4ac = 0, the square root = 0 which gives 1 answer.
•If b^2 - 4ac > 0, the square root > 0. So + or - square root. (2 answers.)
•If b^2 - 4ac < 0, the square root < 0 (negative). And you can't square a negative number. (0 or infinate answers). (In further advanced maths, you theoretically can get an answer in terms of i, but that's an entire topic by itself).
If it has equal roots, (e.g x=2, x=2), it means there is only one value for x (x=2). This means the determinant: b^2 - 4ac = 0.
ax^2+bx+c=0
x={-b±√(b^2–4ac)}/2a
as per condition
{-b+√(b^2–4ac)}/2a={-b-√(b^2–4ac)}/2a
or, -b+√(b^2–4ac)=-b-√(b^2–4ac)
or, √(b^2–4ac)=-√(b^2–4ac)
or, 2√(b^2–4ac)=0
or b^2=4ac
now x ={-b+√(b^2–4ac)}/2a
={-b+√(b^2-b^2)}2a
=-b/2a
equal root is -b/2a
Two roots will be equal if the graph touches the x-axis and value of expression will be minimum at the vertex (minimum ) of graph formed by equation ax² + bx + c = 0.
For Maxima or minima, derivative of expression = 0.
=> dy/dx = 2ax + b = 0
=> x = -b/2a.
:-)
Hope this will help you…