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Difficulty: MediumQuadratic Equations

If the quadratic equation x2−bx+16=0x^2 - bx + 16 = 0, where bb is a positive constant, has two real solutions such that one solution is 44 times the other, what is the value of bb?

Answer: 10

Answer

The value of bb is 1010.
By writing the roots as rr and 4r4r, the quadratic equation can be represented as (x−r)(x−4r)=x2−5rx+4r2=0(x-r)(x-4r) = x^2 - 5rx + 4r^2 = 0. Comparing this to the given equation x2−bx+16=0x^2 - bx + 16 = 0, we establish that 4r2=164r^2 = 16 and b=5rb = 5r. Solving for rr gives r2=4r^2 = 4, which means r=±2r = \pm 2. Since bb is a positive constant, we select r=2r = 2, yielding b=5(2)=10b = 5(2) = 10. Alternatively, using Vieta's formulas, the product of the roots is r⋅4r=16  ⟹  4r2=16  ⟹  r=±2r \cdot 4r = 16 \implies 4r^2 = 16 \implies r = \pm 2, and the sum of the roots is r+4r=b  ⟹  5r=br + 4r = b \implies 5r = b. Since b>0b > 0, we find b=10b = 10.

Step-by-Step Solution

1
Define the roots in terms of a single variable and express the quadratic equation in factored form.
Let the roots be rr and 4r4r. The factored form is (x−r)(x−4r)=0(x - r)(x - 4r) = 0.
This allows us to relate the given relationship between the roots to the coefficients of the quadratic equation.
2
Expand the factored quadratic expression.
x2−5rx+4r2=0x^2 - 5rx + 4r^2 = 0
This transforms the equation into the standard form x2+Bx+C=0x^2 + Bx + C = 0 so we can match coefficients.
3
Compare the expanded equation with the given equation x2−bx+16=0x^2 - bx + 16 = 0.
4r2=164r^2 = 16 and b=5rb = 5r
Matching corresponding coefficients allows us to solve for the unknown variables.
4
Solve the constant term equation for rr.
r=2r = 2 or r=−2r = -2
Finding the value of rr is necessary to calculate the value of bb.
5
Calculate the value of bb using the constraint that bb is positive.
b=5(2)=10b = 5(2) = 10
Using the negative root r=−2r = -2 would yield a negative value for bb (b=−10b = -10), which violates the constraint that bb must be a positive constant.

Key Concept

Relating the roots of a quadratic equation to its coefficients
Estimated Time:1m 30s
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Topics

Quadratic Equations

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