Question

The number of real/solutiôns of the equation \(2|x|^{2}-5|x|+2=0\) is: (a) 0 (b) 4 (c) 2 (d) none of thes:

Short answer

Answer: The equation has 4 real solutions.

Step-by-step solution

Rewrite the equation as a quadratic equation in |x|.

Let \(y = |x|\). The equation then becomes: \(2y^2 - 5y + 2 = 0\)

Solve the quadratic equation for y.

To find the solutions for y, we can try factoring the quadratic, or using the quadratic formula if needed. In this case, we can factor the equation as follows: \((2y-1)(y-2) = 0\) Now, we have two possible solutions for y: \(2y - 1 = 0 \Rightarrow y = \frac{1}{2}\) \(y - 2 = 0 \Rightarrow y = 2\)

Find the real solutions for x.

Remember that \(y = |x|\). For each value of y, we can find the corresponding values of x: If \(y = \frac{1}{2}\), \(x = \frac{1}{2}\) or \(x = -\frac{1}{2}\) If \(y = 2\), \(x = 2\) or \(x = -2\)

Count the number of distinct real solutions.

The four solutions found for x are distinct. Therefore, the given equation has 4 real solutions. The correct option is (b) 4.

Key concepts

Understanding Absolute Value Equations

Absolute value equations are special because they involve the absolute value of a variable or expression, represented as \(|x|\). Absolute value measures the distance of a number from zero on the number line, always being non-negative. In our exercise, the equation \(2|x|^{2}-5|x|+2=0\) is given.

This equation involves \(|x|\) rather than \(x\), which means that no matter if \(x\) is positive or negative, the absolute value remains positive. To simplify such equations, we often substitute \(y = |x|\) as done in the provided solution. This transforms it into a more straightforward quadratic equation \(2y^2 - 5y + 2 = 0\), avoiding the absolute value complication.

Once this substitution is made, we solve for \(y\), and later translate those solutions back to \(x\) by considering \(x = y\) and \(x = -y\). This is because both positive and negative values for \(x\) give the same absolute value.

Factoring Quadratics to Find Solutions

Factoring quadratics is a key method to solve quadratic equations like \(2y^2 - 5y + 2 = 0\). Here, the quadratic can be expressed as the product of two binomials: \((2y-1)(y-2)=0\). This factored form allows us to solve for individual values of \(y\) that make the equation true.

To factor a quadratic, look for two numbers that multiply to the constant term (here 2) and add up to the coefficient of the linear term (here -5).

We can check the factors by multiplying them back to ensure they give the original quadratic equation. After factoring, set each factor equal to zero to solve for \(y\):

• \(2y-1=0\) gives \({y= \frac{1}{2}}\)
• \(y-2=0\) gives \({y = 2}\)

These are the potential values for the absolute value of \(x\).

Identifying Real Solutions in Quadratic Equations

Real solutions in quadratic equations are values for the variable that satisfy the equation. When we solve \(2y^2 - 5y + 2 = 0\), finding \(y = \frac{1}{2}\) and \(y = 2\) indicates the absolute values \(|x|\) can take on these two real values.

For each solution to \(y\), we consider both positive and negative forms to convert back to \(x\):

• If \(y = \frac{1}{2}\), then \(x = \frac{1}{2}\) or \(x = -\frac{1}{2}\)
• If \(y = 2\), then \(x = 2\) or \(x = -2\)

Real solutions are the actual answers we can verify by substituting them back into the original equation.

Real solutions are useful for determining potential outcomes in practical scenarios, like physics or engineering problems.

Counting Distinct Solutions in Algebra

In algebra, distinct solutions mean different values that make the equation true. For \(2|x|^{2}-5|x|+2=0\), we ultimately find four distinct real solutions: \({x = \frac{1}{2}, -\frac{1}{2}, 2, -2}\).

Counting distinct solutions involves ensuring none are repeated, and all satisfy the original equation. Each \(x\) value derived corresponds back to a unique solution for the quadratic transformed from the absolute value equation.

Understanding distinct solutions helps in analyzing systems where unique outputs are essential, such as choosing configurations in technology or solving multi-variable systems.

By confirming each solution is valid, we see why the answer to the exercise is 4 distinct solutions. This approach ensures broad coverage of all potential answers.