One of the solutions to the quadratic equation 0.5x^2 + bx - 6 = 0, where b is a constant,… — ACT

One of the solutions to the quadratic equation $0.5x^2 + bx - 6 = 0$ , where $b$ is a constant, is $x = 3$ . What is the value of the other solution?

Answer: -4

Answer

The other solution to the quadratic equation is -4.

Substituting the given solution

x = 3

into the equation yields

0.5(3)^2 + 3b - 6 = 0

. Simplifying this expression gives

4.5 + 3b - 6 = 0

, which leads to

3b = 1.5

and thus

b = 0.5

. With

b = 0.5

, the quadratic equation becomes

0.5x^2 + 0.5x - 6 = 0

. Multiplying the entire equation by 2 to obtain integer coefficients results in

x^2 + x - 12 = 0

. This quadratic factors into

(x - 3)(x + 4) = 0

, which gives the solutions

x = 3

and

x = -4

. Therefore, the other solution is

-4

. Alternatively, using Vieta's formulas, the product of the roots of a quadratic equation

ax^2 + bx + c = 0

is equal to

c/a

. Here, the product of the roots is

-6 / 0.5 = -12

. Since one root is

3

, the other root must be

-12 / 3 = -4

Step-by-Step Solution

Substitute the given solution

x = 3

into the quadratic equation to find the value of

b

b = 0.5

Since

x = 3

is a solution, it must satisfy the equation, allowing us to solve for the unknown coefficient

b

Rewrite the equation using

b = 0.5

and simplify by multiplying all terms by 2.

x^2 + x - 12 = 0

Multiplying the equation by 2 eliminates the decimal coefficients, making the quadratic expression easier to factor.

Factor the quadratic equation to determine the roots.

x = 3

x = -4

The equation factors into

(x - 3)(x + 4) = 0

. Solving for

x

yields the given root of 3 and the second root of -4.

Key Concept

Solving quadratic equations by utilizing a known solution to determine unknown coefficients, and applying factoring techniques or root relationships to find the remaining solution.

Estimated Time:1m 30s

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