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Question

Difficulty: MediumQuadratic Equations

In the quadratic equation x2−6x−k=0x^2 - 6x - k = 0, kk is a positive constant. If the solutions to the equation are x=3±17x = 3 \pm \sqrt{17}, what is the value of kk?

Answer: 8

Answer

8
By applying the quadratic formula to x2−6x−k=0x^2 - 6x - k = 0, we find the solutions are x=3±9+kx = 3 \pm \sqrt{9 + k}. Equating the expression inside the radical to the given solutions 3±173 \pm \sqrt{17} yields 9+k=179 + k = 17. Solving for kk gives 8.

Step-by-Step Solution

1
Identify the coefficients of the quadratic equation.
a=1a = 1, b=−6b = -6, and c=−kc = -k
To apply the quadratic formula, we need to know the values of aa, bb, and cc from the standard form ax2+bx+c=0ax^2 + bx + c = 0.
2
Substitute the coefficients into the quadratic formula.
x=6±(−6)2−4(1)(−k)2(1)x = \frac{6 \pm \sqrt{(-6)^2 - 4(1)(-k)}}{2(1)}
The quadratic formula x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} gives the solutions directly.
3
Simplify the expression inside the radical and divide by 2.
x=3±9+kx = 3 \pm \sqrt{9 + k}
Simplifying the radical expression allows us to easily compare it to the given solutions.
4
Compare the simplified solution to the given solution form.
9+k=179 + k = 17
Since the solutions are given as 3±173 \pm \sqrt{17}, the expression inside the radical must equal 17.
5
Solve for the constant kk.
k=8k = 8
Subtracting 9 from both sides of the equation yields the value of the constant.

Key Concept

Solving quadratic equations using the quadratic formula or by completing the square to find the relationship between coefficients and solutions.

Alternative Method

Instead of using the quadratic formula, the equation can be solved by completing the square. Rewrite the equation as x2−6x=kx^2 - 6x = k. Adding 9 to both sides gives x2−6x+9=k+9x^2 - 6x + 9 = k + 9, which can be factored as (x−3)2=k+9(x - 3)^2 = k + 9. Taking the square root of both sides yields x=3±k+9x = 3 \pm \sqrt{k + 9}. Comparing this to the given solutions 3±173 \pm \sqrt{17}, we get k+9=17k + 9 = 17, so k=8k = 8.
Estimated Time:1m 30s
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Topics

Quadratic Equations

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