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Chapter 8: Problem 39

Solve each equation. Check the solutions. $$x=\sqrt{7 x-10}$$

Short Answer

Expert verified
Solutions are x = 2 and x = 5.

Step by step solution

01

- Square both sides

To remove the square root, square both sides of the equation. This gives:\[x^2 = (\frac{7}{x} - 10)^2\]
02

- Expand the right side

Expand the right side of the equation:\[x^2 = 7 x - 10\]
03

- Rearrange the equation

Move all terms to one side to set the equation to zero:\[x^2 - 7 x + 10 = 0\]
04

- Factorize the quadratic equation

Factor the quadratic equation. The factored form is:\[(x-2)(x-5) = 0\]
05

- Solve for x

Set each factor to zero and solve for x:\[x - 2 = 0 \Rightarrow x = 2\]\[x - 5 = 0 \Rightarrow x = 5\]
06

- Check the solutions

Substitute x = 2 and x = 5 back into the original equation to verify the solutions:For \(x = 2\):\[2 = \sqrt{7(2) - 10}\Rightarrow 2 = \sqrt{4}\Rightarrow 2 = 2\] (True)For \(x = 5\):\[5 = \sqrt{7(5) - 10}\Rightarrow 5 = \sqrt{25}\Rightarrow 5 = 5\] (True)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Square Both Sides
Succeeding in solving quadratic equations often starts with eliminating square roots. When faced with an equation like: \[x = \sqrt{7x - 10}\], we can remove the square root by squaring both sides. This turns the original equation into:\[x^2 = (7x - 10)\].This is vital because squaring both sides simplifies the equation, helping us isolate and manage x more easily. Remember, when you square a square root, you effectively cancel out the square root.
Factoring Quadratic Equations
Factoring a quadratic equation is an essential skill. When your equation is in the form\[x^2 - 7x + 10 = 0\], the goal is to express it as a product of binomials. This is done as follows:\[(x - 2)(x - 5) = 0\].Each binomial represents a possible value of x. Start by finding two numbers that multiply to give you the constant term (10 in this case) and add to give you the middle coefficient (-7 here). This helps break down complex quadratics into manageable pieces.
Checking Solutions
Always verify your solutions. After solving for x, substitute the values back into the original equation to ensure they work. For instance, if we have \(x = 2\) and \(x = 5\) as solutions, we substitute them back: For \(x = 2\):\[2 = \sqrt{7(2) - 10}\Rightarrow 2 = \sqrt{4}\Rightarrow 2 = 2\], which is true. Similarly for \(x = 5\):\[5 = \sqrt{7(5) - 10}\Rightarrow 5 = \sqrt{25}\Rightarrow 5 = 5\], also true. This ensures our original equation holds, confirming the solutions are correct.

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