Question

solve for x, where x is a real number.\\(\sqrt{7x - 10} = x\\)\
if there is more than one solution, separate them with commas.\
if there is no solution, click on
o solution\.\
x = \boxed{}

Explanation:

Step1: Square both sides to eliminate the square root

To solve the equation \(\sqrt{7x - 10}=x\), we first square both sides. Squaring the left side gives us \(7x - 10\) (since \((\sqrt{a})^2=a\) for \(a\geq0\)), and squaring the right side gives us \(x^2\). So we get the equation:

$$x^2 = 7x - 10$$

Step2: Rearrange into standard quadratic form

We rearrange the equation \(x^2 = 7x - 10\) to the standard quadratic form \(ax^2+bx + c = 0\) by subtracting \(7x\) and adding \(10\) to both sides:

$$x^2-7x + 10=0$$

Step3: Factor the quadratic equation

We factor the quadratic \(x^2-7x + 10\). We need two numbers that multiply to \(10\) and add up to \(- 7\). The numbers are \(-2\) and \(-5\). So we can factor the quadratic as:

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

Step4: Solve for x using the zero - product property

The zero - product property states that if \(ab = 0\), then either \(a = 0\) or \(b = 0\). So for \((x - 2)(x - 5)=0\), we have two cases:

• Case 1: \(x - 2=0\), which gives \(x = 2\)
• Case 2: \(x - 5=0\), which gives \(x = 5\)

Step5: Check the solutions in the original equation

• For \(x = 2\): Substitute \(x = 2\) into the original equation \(\sqrt{7x-10}\). We get \(\sqrt{7\times2 - 10}=\sqrt{14 - 10}=\sqrt{4}=2\), and the right side is \(x = 2\). So \(x = 2\) is a valid solution.
• For \(x = 5\): Substitute \(x = 5\) into the original equation \(\sqrt{7x-10}\). We get \(\sqrt{7\times5 - 10}=\sqrt{35 - 10}=\sqrt{25}=5\), and the right side is \(x = 5\). So \(x = 5\) is a valid solution.

Answer:

2, 5