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Difficulty: MediumQuadratic Equations

In the quadratic equation x2−8x+k=0x^2 - 8x + k = 0, kk is a constant. If the difference between the two real solutions to the equation is 22, what is the value of kk?

  1. 15Answer
  2. B
    -15
  3. C
    12
  4. D
    8

Answer

15
The correct answer is 15. The sum of the solutions to the quadratic equation x2−8x+k=0x^2 - 8x + k = 0 is given by −−81=8-\frac{-8}{1} = 8. We are given that the difference between the two solutions is 22. Let the two solutions be x1x_1 and x2x_2. We can set up the system of equations: x1+x2=8x_1 + x_2 = 8 and x1−x2=2x_1 - x_2 = 2. Adding the equations gives 2x1=102x_1 = 10, so x1=5x_1 = 5. Thus, x2=3x_2 = 3. The product of the solutions is equal to the constant term kk. Therefore, k=x1×x2=5×3=15k = x_1 \times x_2 = 5 \times 3 = 15. Alternatively, using the difference of roots formula, b2−4aca=2⇒(−8)2−4(1)(k)1=2⇒64−4k=2\frac{\sqrt{b^2 - 4ac}}{a} = 2 \Rightarrow \frac{\sqrt{(-8)^2 - 4(1)(k)}}{1} = 2 \Rightarrow \sqrt{64 - 4k} = 2. Squaring both sides gives 64−4k=4⇒4k=60⇒k=1564 - 4k = 4 \Rightarrow 4k = 60 \Rightarrow k = 15.

Step-by-Step Solution

1
Find the sum of the two solutions of the quadratic equation.
The sum of the solutions is 88.
For a quadratic equation in the form x2+bx+c=0x^2 + bx + c = 0, the sum of the solutions is equal to −b-b. Here, −(−8)=8-(-8) = 8.
2
Set up a system of linear equations using the sum and the given difference of the solutions to find the individual solutions.
The solutions are 55 and 33.
Let the solutions be x1x_1 and x2x_2. We have the system: x1+x2=8x_1 + x_2 = 8 and x1−x2=2x_1 - x_2 = 2. Adding these two equations gives 2x1=102x_1 = 10, so x1=5x_1 = 5. Substituting 55 back into the first equation gives 5+x2=85 + x_2 = 8, so x2=3x_2 = 3.
3
Use the product of the solutions to determine the value of kk.
The value of kk is 1515.
For a quadratic equation in the form x2+bx+c=0x^2 + bx + c = 0, the product of the solutions is equal to cc, which is kk. Therefore, k=x1×x2=5×3=15k = x_1 \times x_2 = 5 \times 3 = 15.

Key Concept

Relationship between the roots and coefficients of a quadratic equation (Vieta's Formulas).
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Topics

Quadratic Equations

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