What is the sum of all real solutions to the equation x(x - 3) = 10? — GMAT

What is the sum of all real solutions to the equation $x(x - 3) = 10$ ?

The sum of all real solutions to the equation is 3.

Expanding the equation

x(x - 3) = 10

yields

x^2 - 3x - 10 = 0

. Factoring this expression gives

(x - 5)(x + 2) = 0

, yielding solutions

x = 5

and

x = -2

. The sum of these two real solutions is

5 + (-2) = 3

. Alternatively, applying Vieta's formula directly to

x^2 - 3x - 10 = 0

gives the sum of roots as

-\frac{b}{a} = -\frac{-3}{1} = 3

Expand the left side of the equation.

x^2 - 3x = 10

Multiply

x

across the binomial

(x - 3)

to convert to standard polynomial form.

Rearrange the equation into standard quadratic form

ax^2 + bx + c = 0

x^2 - 3x - 10 = 0

Subtract 10 from both sides so the equation equals zero.

Factor the quadratic expression.

(x - 5)(x + 2) = 0

Find two numbers that multiply to

-10

and add up to

-3

, which are

-5

and

2

Solve for the roots of the equation.

x = 5

x = -2

Apply the zero-product property: set each factor equal to zero.

Calculate the sum of the solutions.

Sum

= 5 + (-2) = 3

Add the two real solutions together. Alternatively, by Vieta's formulas, the sum of roots is

-\frac{b}{a} = -\frac{-3}{1} = 3

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