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Problem 477

Solve the equation $$2 \mathrm{x}^{2}-5 \mathrm{x}+3=0$$

Expert verified
The solutions to the quadratic equation $$2x^2 - 5x + 3 = 0$$ are $$x_1 = \frac{3}{2}$$ and $$x_2 = 1$$.
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Step 1: Identify the coefficients a, b, and c

In the given quadratic equation $$2x^2 - 5x + 3 = 0$$, we can identify the coefficients as: - $$a = 2$$ - $$b = -5$$ - $$c = 3$$

Step 2: Apply the quadratic formula

Recall, the quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.$ Plug in the values of a, b, and c into the formula and simplify. $x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(2)(3)}}{2(2)}$ $x = \frac{5 \pm \sqrt{25 - 24}}{4}$

Step 3: Calculate the discriminant and simplify

The discriminant is the expression inside the square root, $$b^2 - 4ac$$. In our case, the discriminant is $$25 - 24 = 1$$. We will now simplify the quadratic formula: $x = \frac{5 \pm \sqrt{1}}{4}$

Step 4: Calculate the values of x

Since the square root of 1 is 1, we can find the two possible values for x: $x_1 = \frac{5 + 1}{4} = \frac{6}{4} = \frac{3}{2}$ $x_2 = \frac{5 - 1}{4} = \frac{4}{4} = 1$ Therefore, the solutions to the quadratic equation $$2x^2 - 5x + 3 = 0$$ are $$x_1 = \frac{3}{2}$$ and $$x_2 = 1$$.

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