In the quadratic equation x^2 - 10x + c = 0, c is a constant. If the two real solutions to… — SAT

In the quadratic equation $x^2 - 10x + c = 0$ , $c$ is a constant. If the two real solutions to the equation have a difference of 6, what is the value of $c$ ?

Answer: 16

Answer

The correct answer is 16. By using the quadratic formula, the two solutions of the equation

x^2 - 10x + c = 0

are

5 + \sqrt{25 - c}

and

5 - \sqrt{25 - c}

. The difference between these two solutions is

2\sqrt{25 - c}

. Given that the difference is 6, we set

2\sqrt{25 - c} = 6

, which simplifies to

\sqrt{25 - c} = 3

. Squaring both sides gives

25 - c = 9

, which yields

c = 16

Step-by-Step Solution

Use the quadratic formula to express the solutions of

x^2 - 10x + c = 0

The solutions are

x = 5 \pm \sqrt{25 - c}

This expresses the roots of the quadratic equation in terms of the constant

c

Set the difference between the two solutions equal to 6.

(5 + \sqrt{25 - c}) - (5 - \sqrt{25 - c}) = 6

, which simplifies to

2\sqrt{25 - c} = 6

We are given that the two real solutions have a difference of 6.

Solve the equation

2\sqrt{25 - c} = 6

for

c

\sqrt{25 - c} = 3 \implies 25 - c = 9 \implies c = 16

This isolates the constant

c

using standard algebraic operations.

Key Concept

Solving quadratic equations and using properties of roots.

Alternative Method

Alternatively, we can use the relationship between the roots of a quadratic equation. If the roots are

x_1

and

x_2

, then

x_1 + x_2 = 10

and

x_1 x_2 = c

. Using the identity

(x_1 - x_2)^2 = (x_1 + x_2)^2 - 4x_1 x_2

, we substitute the given values:

(6)^2 = (10)^2 - 4c

. This simplifies to

36 = 100 - 4c

, which gives

4c = 64

, or

c = 16

Estimated Time:1m 30s

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Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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