One of the solutions to the quadratic equation x^2 + 8x + c = 0, where c is a constant, is… — SAT

One of the solutions to the quadratic equation $x^2 + 8x + c = 0$ , where $c$ is a constant, is $x = -4 + \sqrt{5}$ . What is the value of $c$ ?

The value of 11 is correct because completing the square of the equation

x^2 + 8x + c = 0

yields

(x + 4)^2 = 16 - c

, which gives solutions of the form

x = -4 \pm \sqrt{16 - c}

. Comparing this to the given solution

x = -4 + \sqrt{5}

shows that

16 - c = 5

, which simplifies to

c = 11

. Direct substitution of the solution back into the equation also confirms this result.

Express the quadratic equation in vertex form by completing the square.

x^2 + 8x + c = (x + 4)^2 - 16 + c = 0

Completing the square allows us to easily solve for the variable in terms of the constant.

Isolate the squared term and solve for the variable.

(x + 4)^2 = 16 - c \implies x = -4 \pm \sqrt{16 - c}

Taking the square root of both sides gives the general solutions of the quadratic equation.

Equate the general solution to the given solution to find the value of the constant.

Since one solution is

-4 + \sqrt{5}

, we set

16 - c = 5

, which gives

c = 11

Matching the terms under the radical determines the value of the constant.

Solving quadratic equations by completing the square or using the quadratic formula

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