Question

solve the radical equation. check for extraneous solutions.
$2x = \sqrt{17x - 15}$

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. $x = \square$
(type your answer(s) as integers or simplified fractions. use a comma to separate answers as needed.)
b. there are no solutions.

Explanation:

Step1: Square both sides to eliminate the square root

To solve the radical equation \(2x = \sqrt{17x - 15}\), we first square both sides of the equation. Squaring the left side gives \((2x)^2 = 4x^2\), and squaring the right side gives \((\sqrt{17x - 15})^2 = 17x - 15\). So we have the quadratic equation:
\[4x^2 = 17x - 15\]

Step2: Rearrange into standard quadratic form

We rearrange the equation \(4x^2 = 17x - 15\) to the standard quadratic form \(ax^2 + bx + c = 0\) by subtracting \(17x\) and adding \(15\) to both sides:
\[4x^2 - 17x + 15 = 0\]

Step3: Factor the quadratic equation

We factor the quadratic \(4x^2 - 17x + 15\). We need two numbers that multiply to \(4\times15 = 60\) and add up to \(-17\). The numbers \(-5\) and \(-12\) work. So we can rewrite the middle term:
\[4x^2 - 12x - 5x + 15 = 0\]
Group the terms:
\[(4x^2 - 12x) + (-5x + 15) = 0\]
Factor out the common factors from each group:
\[4x(x - 3) - 5(x - 3) = 0\]
Now we can factor out \((x - 3)\):
\[(4x - 5)(x - 3) = 0\]

Step4: Solve for x using the zero - product property

Using the zero - product property, if \((4x - 5)(x - 3)=0\), then either \(4x - 5 = 0\) or \(x - 3 = 0\).

• For \(4x - 5 = 0\), add \(5\) to both sides: \(4x=5\), then divide by \(4\): \(x=\frac{5}{4}\)
• For \(x - 3 = 0\), add \(3\) to both sides: \(x = 3\)

Step5: Check for extraneous solutions

We need to check these solutions in the original equation \(2x=\sqrt{17x - 15}\) because squaring both sides can introduce extraneous solutions.

• For \(x = 3\):

Left side: \(2x=2\times3 = 6\)
Right side: \(\sqrt{17\times3 - 15}=\sqrt{51 - 15}=\sqrt{36}=6\)
Since \(6 = 6\), \(x = 3\) is a valid solution.

• For \(x=\frac{5}{4}\):

Left side: \(2x=2\times\frac{5}{4}=\frac{5}{2}\)
Right side: \(\sqrt{17\times\frac{5}{4}-15}=\sqrt{\frac{85}{4}-15}=\sqrt{\frac{85 - 60}{4}}=\sqrt{\frac{25}{4}}=\frac{5}{2}\)
Since \(\frac{5}{2}=\frac{5}{2}\), \(x = \frac{5}{4}\) is also a valid solution.

Answer:

A. \(x=\frac{5}{4},3\)