QuestionQuestion
Solve for all possible values of x . $\sqrt{5 x+26}=x+6$
Answer Altempt 1 out of 2

Studdy Solution

STEP 1

1. The equation involves a square root and a linear expression.
2. We assume $x$ is a real number.
3. We will square both sides to eliminate the square root, but we must check for extraneous solutions.

STEP 2

1. Eliminate the square root by squaring both sides of the equation.
2. Simplify and solve the resulting quadratic equation.
3. Verify solutions to ensure they satisfy the original equation.

STEP 3

Square both sides of the equation to eliminate the square root:
$\sqrt{5x + 26} = x + 6$
Squaring both sides gives:
$(\sqrt{5x + 26})^2 = (x + 6)^2$
This simplifies to:
$5x + 26 = (x + 6)^2$

STEP 4

Expand the right side of the equation:
$5x + 26 = (x + 6)(x + 6)$ $5x + 26 = x^2 + 12x + 36$
Rearrange the equation to set it to zero:
$x^2 + 12x + 36 - 5x - 26 = 0$ $x^2 + 7x + 10 = 0$

STEP 5

Solve the quadratic equation using the quadratic formula or factoring. Here, we will factor:
$x^2 + 7x + 10 = 0$
The factors of 10 that add up to 7 are 5 and 2:
$(x + 5)(x + 2) = 0$
Set each factor to zero:
$x + 5 = 0 \quad \text{or} \quad x + 2 = 0$
Solve for $x$ :
$x = -5 \quad \text{or} \quad x = -2$

STEP 6

Verify each solution by substituting back into the original equation:
For $x = -5$ :
$\sqrt{5(-5) + 26} = -5 + 6$ $\sqrt{-25 + 26} = 1$ $\sqrt{1} = 1$ This solution is valid.
For $x = -2$ :
$\sqrt{5(-2) + 26} = -2 + 6$ $\sqrt{-10 + 26} = 4$ $\sqrt{16} = 4$ This solution is valid.
The possible values of $x$ are:
$\boxed{-5} \quad \text{and} \quad \boxed{-2}$

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