In the equation x(x - 8) = k, k is a constant. If the product of the two real solutions to… — SAT

In the equation $x(x - 8) = k$ , $k$ is a constant. If the product of the two real solutions to the equation is $-20$ , what is the value of the larger solution?

Answer: 10

Answer

The larger solution to the equation is 10.

To find the larger solution, the equation is first rewritten in standard form as

x^2 - 8x - k = 0

. The product of the roots of a quadratic equation in the form

ax^2 + bx + c = 0

\frac{c}{a}

. Here,

a = 1

and

c = -k

, so the product of the roots is

-k

. Given that the product of the roots is

-20

, we can set up the equation

-k = -20

, which gives

k = 20

. Substituting

k = 20

back into the equation yields

x^2 - 8x - 20 = 0

. Factoring the quadratic expression gives

(x - 10)(x + 2) = 0

, which has the solutions

x = 10

and

x = -2

. The larger of these two solutions is 10.

Step-by-Step Solution

Rewrite the given equation

x(x - 8) = k

in standard quadratic form

ax^2 + bx + c = 0

x^2 - 8x - k = 0

To apply quadratic properties, the equation must be in standard form.

Use the product of roots formula to find the value of the constant

k

-k = -20

, which simplifies to

k = 20

The product of the roots of a quadratic equation in standard form is the constant term divided by the leading coefficient.

Substitute the value of

k

back into the quadratic equation and factor it to find the two solutions.

(x - 10)(x + 2) = 0

, so

x = 10

x = -2

Factoring the quadratic equation allows us to find the individual roots.

Compare the two solutions to identify the larger value.

Comparing 10 and -2, 10 is the greater value.

Key Concept

Using the relationship between coefficients and the product of roots to solve a quadratic equation.

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