Question

question
solve for all possible values of x.
\\(\sqrt{8x + 65} = x + 10\\)
answer attempt 1 out of 2
\\(\oplus\\) additional solution \\(\circleddash\\) no solution
\\(x = \square\\) submit answer

Explanation:

Step1: Square both sides to eliminate square root

To solve the equation \(\sqrt{8x + 65}=x + 10\), we first square both sides of the equation. Squaring the left side \((\sqrt{8x + 65})^2\) gives \(8x + 65\), and squaring the right side \((x + 10)^2\) using the formula \((a + b)^2=a^2+2ab + b^2\) (where \(a = x\) and \(b = 10\)) gives \(x^{2}+20x + 100\). So we get the new equation:

$$8x+65=x^{2}+20x + 100$$

Step2: Rearrange into quadratic form

Subtract \(8x\) and \(65\) from both sides to set the equation to zero:

$$x^{2}+20x+100 - 8x - 65=0$$

Simplify the like terms: \(20x-8x = 12x\) and \(100 - 65=35\), so the equation becomes:

$$x^{2}+12x + 35=0$$

Step3: Factor the quadratic equation

We need to find two numbers that multiply to \(35\) and add up to \(12\). The numbers \(5\) and \(7\) work because \(5\times7 = 35\) and \(5 + 7=12\). So we can factor the quadratic as:

$$(x + 5)(x+7)=0$$

Step4: Solve for x using zero - product property

The zero - product property states that if \(ab = 0\), then either \(a = 0\) or \(b = 0\). So we set each factor equal to zero:

• If \(x + 5=0\), then \(x=-5\).
• If \(x + 7=0\), then \(x=-7\).

Step5: Check the solutions in the original equation

• For \(x=-5\):

Left side: \(\sqrt{8\times(-5)+65}=\sqrt{-40 + 65}=\sqrt{25}=5\)
Right side: \(-5 + 10 = 5\)
Since the left side equals the right side, \(x = - 5\) is a valid solution.

• For \(x=-7\):

Left side: \(\sqrt{8\times(-7)+65}=\sqrt{-56 + 65}=\sqrt{9}=3\)
Right side: \(-7+10 = 3\)
Since the left side equals the right side, \(x=-7\) is also a valid solution.

Answer:

\(x=-5\) or \(x = - 7\)