What is the sum of all real values of x that satisfy the equation (x - 3)^2 + x(x + 2) = 1… — ACT

What is the sum of all real values of $x$ that satisfy the equation $(x - 3)^2 + x(x + 2) = 15$ ?

The sum of all real values of

x

that satisfy the equation is

2

Expanding the equation yields

2x^2 - 4x + 9 = 15

. Setting this to zero gives

2x^2 - 4x - 6 = 0

. Dividing by the common factor of

2

simplifies this to

x^2 - 2x - 3 = 0

. Factoring the trinomial yields

(x - 3)(x + 1) = 0

, which gives the two solutions

x = 3

and

x = -1

. Summing these two solutions gives

3 + (-1) = 2

Expand both terms on the left side of the equation.

(x^2 - 6x + 9) + (x^2 + 2x) = 15

Applying the binomial squaring formula

(a - b)^2 = a^2 - 2ab + b^2

(x - 3)^2

and distributing

x

to both terms in

x(x + 2)

allows us to simplify the equation.

Combine like terms and set the quadratic equation to zero.

2x^2 - 4x - 6 = 0

Grouping

x^2

terms,

x

terms, and constant terms on one side is necessary to format the quadratic equation as

ax^2 + bx + c = 0

before factoring.

Divide the entire equation by the common factor of

2

to simplify factoring.

x^2 - 2x - 3 = 0

Simplifying the quadratic equation makes it easier to find two binomial factors.

Factor the quadratic trinomial by finding two numbers that multiply to

-3

and add to

-2

(x - 3)(x + 1) = 0

Since

-3 \times 1 = -3

and

-3 + 1 = -2

, we can write the quadratic in factored form.

Set each factor to zero to solve for

x

x = 3

x = -1

Applying the zero product property determines the two values of

x

that satisfy the original equation.

Calculate the sum of the two solutions.

3 + (-1) = 2

The question asks for the sum of all real values of

x

that satisfy the equation.

Solving quadratic equations by rearranging terms, factoring trinomials, and applying the zero product property.

Soru