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Difficulty: MediumQuadratic Equations and the Quadratic Formula

A quadratic equation is defined by x2−bx+18=0x^2 - bx + 18 = 0, where bb is a positive constant. If the difference between the two real solutions of this equation is 3, what is the value of bb?

Answer: 9

Answer

The value of the positive constant bb is 9.
By applying the quadratic formula, the roots of the equation are x=b±b2−722x = \frac{b \pm \sqrt{b^2 - 72}}{2}. The difference between these roots is b2−72\sqrt{b^2 - 72}. Setting this equal to 3 gives b2−72=3\sqrt{b^2 - 72} = 3. Squaring both sides yields b2−72=9b^2 - 72 = 9, which simplifies to b2=81b^2 = 81. Taking the positive root since bb is a positive constant gives b=9b = 9.

Step-by-Step Solution

1
Express the roots of the quadratic equation x2−bx+18=0x^2 - bx + 18 = 0 using the quadratic formula.
The roots are x=b±b2−4(1)(18)2=b±b2−722x = \frac{b \pm \sqrt{b^2 - 4(1)(18)}}{2} = \frac{b \pm \sqrt{b^2 - 72}}{2}.
This provides a formulaic representation of the two solutions in terms of the unknown parameter bb.
2
Subtract the smaller root from the larger root to represent the difference between the solutions, and set this expression equal to 3.
Difference =b+b2−722−b−b2−722=b2−72=3= \frac{b + \sqrt{b^2 - 72}}{2} - \frac{b - \sqrt{b^2 - 72}}{2} = \sqrt{b^2 - 72} = 3.
The problem specifies that the difference between the two real solutions is 3.
3
Square both sides of the equation to eliminate the radical, and solve for the positive constant bb.
b2−72=9⇒b2=81⇒b=9b^2 - 72 = 9 \Rightarrow b^2 = 81 \Rightarrow b = 9 (since bb is positive).
Squaring both sides allows us to isolate b2b^2 and find the value of bb that satisfies the initial condition.

Key Concept

Solving for quadratic coefficients using the difference of roots derived from the quadratic formula.

Alternative Method

Use Vieta's formulas. Let the roots be r1r_1 and r2r_2. We know that r1+r2=br_1 + r_2 = b and r1r2=18r_1 r_2 = 18. We are given that the difference between the roots is 3, so ∣r1−r2∣=3|r_1 - r_2| = 3. We can use the algebraic identity (r1−r2)2=(r1+r2)2−4r1r2(r_1 - r_2)^2 = (r_1 + r_2)^2 - 4r_1 r_2. Substituting the known values gives 32=b2−4(18)3^2 = b^2 - 4(18), which simplifies to 9=b2−729 = b^2 - 72, leading to b2=81b^2 = 81. Since b>0b > 0, we find b=9b = 9.
Estimated Time:1m 30s
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Topics

Quadratic Equations and the Quadratic Formula

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