Find all the real-number roots of each equation. In each...

Find all the real-number roots of each equation. In each case, give an exact expression for the root and also (where appropriate) a calculator approximation rounded to three decimal places.
$$\log _{10}\left(2 x^{2}-3 x\right)=2$$

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Key Concepts

Logarithmic Equations

Logarithmic equations involve logarithms with expressions that include a variable. Solving these equations typically requires manipulating the expressions within the logarithm and often converting the equation into another form to isolate the variable.

Conversion to Exponential Form

A key technique in solving logarithmic equations is rewriting them in exponential form. By using the definition log_b(A) = C, the equation can be transformed into A = b^C. This conversion simplifies the process of finding the variable's value.

Quadratic Equations

Often, after transforming a logarithmic equation into its exponential form, the resulting equation is quadratic in nature. Quadratic equations, which are second-degree polynomial equations, can be solved using methods such as factoring, completing the square, or applying the quadratic formula.

Domain Restrictions

Since logarithmic functions are only defined for positive arguments, it is essential to consider domain restrictions when solving logarithmic equations. This ensures that any proposed solutions do not lead to taking the logarithm of a non-positive number, which would be invalid.

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