+354 Find the constant k such that the quadratic 2x^2 + 3x + 8x - x^2 + 4x + k has a double root.
+953 Let's first combine all like terms and simplify the equation. We have
\(x^{2}+15x+k\)
Now, there are many ways for this equation to be a double root.
As long as two factors of k add up to 15, we are good.
For example, we have
\(k=36; x^2+15x+36 = (x+12)(x+3)\\ k=14; x^2+15x+14 = (x+14)(x+1)\\\)
Negative k values also work. For example. we have
\(k=-54; x^2+15x-54= (x+18)(x-3)\\\)
Thanks! :)
+879
Find the constant k such that the quadratic 2x^2 + 3x + 8x - x^2 + 4x + k has a double root.
2x2 + 3x + 8x – x2 + 4x + k
Combine like terms x2 + 15x + k
In math, the term "double root" means "repeated root" ... i.e., both roots are the same
Examples: x2 + 2x + 1 factors to (x+1)(x+1) and both roots are –1
x2 + 6x + 9 factors to (x+3)(x+3) and both roots are –3
x2 – 8x + 16 factors to (x–4)(x–4) and both roots are +4
One way to determine that a quadratic has a double root is when its discriminant equals zero
The discriminant is the expression underneath the radical in the quadratic equation
When a quadratic is posed as ax2 + bx + c the discriminant is b2 – 4ac
In x2 + 15x +k in this problem, the discriminant is 152 – (4)(1)(k)
Multiply it out, set equal to 0, and solve for k 225 – 4k = 0
–4k = –225
k = 56.25
The problem doesn't ask for the repeated root, but it's the square root of k.
The factors of the quadratic are (x + 7.5)(x + 7.5) and the repeated root is –7.5
check answer
x2 + 15x + 56.25 = 0
(–7.5)2 + (15)(–7.5) + 56.25 = 0
56.25 – 112.5 + 56.25 = 0
0 = 0
.
