Verified Answer :

  • Identify the coefficients of the quadratic equation \(x^2 + 2x - 3 = 0\), which are \(a=1\), \(b=2\), and \(c=-3\).
  • Factor the trinomial into \((x+3)(x-1) = 0\) by finding two numbers that sum to \(2\) and multiply to \(-3\).
  • Solve the resulting linear equations \(x+3=0\) and \(x-1=0\) to find the roots \(x=-3\) and \(x=1\).
  • Verify both values in the original equation, confirming the solutions are \(\boxed{-3, 1}\).

Explanation

  1. Analysis of the Quadratic Equation

The problem asks us to solve the quadratic equation \(x^2 + 2x - 3 = 0\). This equation is in the standard quadratic form \(ax^2 + bx + c = 0\), where \(a=1\), \(b=2\), and \(c=-3\). We can solve this by factoring, completing the square, or using the quadratic formula. Our goal is to identify all real values of \(x\) that satisfy this equation.

  1. Solving via Factoring

We look for two numbers that multiply to \(ac = (1)(-3) = -3\) and add to \(b = 2\). The numbers \(3\) and \(-1\) satisfy these conditions since \(3 \times (-1) = -3\) and \(3 + (-1) = 2\). We can rewrite the equation as \((x+3)(x-1) = 0\). Setting each factor equal to zero gives us the roots \(x+3=0\) or \(x-1=0\), which results in \(x=-3\) and \(x=1\).

  1. Verification and Final Answer

To verify our roots, we substitute them back into the original equation \(x^2 + 2x - 3 = 0\). For \(x=1\): \((1)^2 + 2(1) - 3 = 1 + 2 - 3 = 0\). For \(x=-3\): \((-3)^2 + 2(-3) - 3 = 9 - 6 - 3 = 0\). Both values satisfy the equation. Thus, the solution set is \(\{-3, 1\}\).

Examples

Quadratic equations like \(x^2 + 2x - 3 = 0\) are fundamental in physics and engineering. For example, if you are modeling the trajectory of a ball thrown into the air, the height of the ball over time might be described by a quadratic function. Finding the roots of the equation helps determine when the ball hits the ground, as that is when the height is zero. By solving such equations, you can predict physical outcomes in various real-world scenarios.

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