Find the constant k such that the quadratic 2x^2 + 3x + 8x - x^2 + 4x + k has a double root.

gnistory Jun 27, 2024

#1**0 **

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! :)

NotThatSmart Jun 27, 2024

#2**0 **

*Find the constant k such that the quadratic 2x^2 + 3x + 8x - x^2 + 4x + k has a double root.*

2x^{2} + 3x + 8x – x^{2} + 4x + k

Combine like terms **x ^{2} + 15x + k**

In math, the term "double root" means "repeated root" ... i.e., both roots are the same

Examples: x^{2} + 2x + 1 factors to (x+1)(x+1) and both roots are –1

x^{2} + 6x + 9 factors to (x+3)(x+3) and both roots are –3

x^{2} – 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 ax^{2} + bx + c the discriminant is b^{2} – 4ac

In x^{2} + 15x +k in this problem, the discriminant is 15^{2} – (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

x^{2} + 15x + 56**.**25 = 0

(–7**.**5)^{2} + (15)(–7**.**5) + 56**.**25 = 0

^{ } 56**.**25 – 112**.**5 + 56**.**25 = 0

^{ } 0 = 0

_{.}

Bosco Jun 28, 2024