Verified Answer :

  • Rearrange the equation into standard form: \(2x^2 + 3x - 2 = 0\).
  • Factor the quadratic expression: \((2x - 1)(x + 2) = 0\).
  • Determine the roots: \(x = \frac{1}{2}\) and \(x = -2\).
  • The values are \(\boxed{a = \frac{1}{2}, b = -2}\).

Explanation

  1. Analyzing the Quadratic Equation

The given problem asks us to solve the quadratic equation \(2x^2 = 2 - 3x\). To find the values of \(a\) and \(b\), we first rearrange the equation into the standard quadratic form \(ax^2 + bx + c = 0\). By moving all terms to the left side, we get \(2x^2 + 3x - 2 = 0\). This is a standard quadratic equation where the coefficients are \(a'=2\), \(b'=3\), and \(c'=-2\).

  1. Factoring the Quadratic Expression

To solve the quadratic equation \(2x^2 + 3x - 2 = 0\), we can use the method of factoring. We look for two numbers that multiply to \(a' \times c' = 2 \times (-2) = -4\) and add up to the middle coefficient \(b' = 3\). The numbers \(4\) and \(-1\) satisfy these conditions since \(4 \times (-1) = -4\) and \(4 + (-1) = 3\). Thus, we can rewrite the equation as \(2x^2 + 4x - x - 2 = 0\). Grouping the terms gives \(2x(x + 2) - 1(x + 2) = 0\), which simplifies to \((2x - 1)(x + 2) = 0\).

  1. Solving for the Roots

From the factored form \((2x - 1)(x + 2) = 0\), we can find the roots by setting each factor equal to zero.
Setting \(2x - 1 = 0\) gives \(2x = 1\), so \(x = \frac{1}{2}\).
Setting \(x + 2 = 0\) gives \(x = -2\).
Thus, the values of \(a\) and \(b\) are \(\frac{1}{2}\) and \(-2\). The roots of the equation \(2x^2 + 3x - 2 = 0\) are \(x = \frac{1}{2}\) and \(x = -2\).

Examples

Quadratic equations are widely used in physics and engineering. For example, when you throw a ball, its height over time follows a quadratic function. If you want to know when the ball hits the ground, you set the height to zero and solve the resulting quadratic equation for the time \(t\). This method of factoring or using the quadratic formula allows us to predict such events accurately.

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