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# Algebra

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

Jun 27, 2024

#1
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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! :)

Jun 27, 2024
#2
+944
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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

x2 + 15x + 56.25 = 0

(–7.5)2 + (15)(–7.5) + 56.25 = 0

56.25 – 112.5 + 56.25 = 0

0 = 0

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Jun 28, 2024