Answered

Without solving the equation $x^2+2x=3$, find the product and the sum of the solutions.



The product is [blank] and the sum is [blank].

Verified Answer :

  • Rewrite the equation in standard form as \(x^2 + 2x - 3 = 0\) to identify coefficients \(a = 1\), \(b = 2\), and \(c = -3\).
  • Apply Vieta's formulas to calculate the sum of the roots: \(-\frac{b}{a} = -2\).
  • Apply Vieta's formulas to calculate the product of the roots: \(\frac{c}{a} = -3\).
  • The final product and sum are \(\boxed{-3}\) and \(\boxed{-2}\) respectively.

Explanation

  1. Standardizing the Equation and Identifying Coefficients

The problem asks for the product and the sum of the solutions of the quadratic equation \(x^2 + 2x = 3\) without directly solving for the roots. We start by bringing the equation into its standard form, which is \(ax^2 + bx + c = 0\). By subtracting 3 from both sides, we get:
\[x^2 + 2x - 3 = 0\]
From this equation, we can identify the coefficients:
\(a = 1\)
\(b = 2\)
\(c = -3\)

  1. Applying Vieta's Formulas

Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. For a quadratic equation in the form \(ax^2 + bx + c = 0\), if the roots are \(x_1\) and \(x_2\), then:

  • The sum of the roots is given by \(x_1 + x_2 = -\frac{b}{a}\).
  • The product of the roots is given by \(x_1 \cdot x_2 = \frac{c}{a}\).

Using these formulas allows us to find the required values directly from the coefficients we identified in the previous step.

  1. Calculating the Sum and Product of the Roots

Using the coefficients \(a = 1\), \(b = 2\), and \(c = -3\):

  • The sum of the roots is \(-\frac{b}{a} = -\frac{2}{1} = -2\).
  • The product of the roots is \(\frac{c}{a} = \frac{-3}{1} = -3\).

Thus, the product of the solutions is \(-3\) and the sum of the solutions is \(-2\).

Examples

Understanding the relationship between coefficients and roots of equations is essential in fields like physics and engineering. For example, when analyzing the motion of an object (like a projectile) defined by a quadratic displacement equation, the sum and product of the roots can help identify total time intervals or equilibrium points without needing to calculate the precise motion at every second. This shortcut saves time and provides a deeper insight into the underlying properties of the mathematical model.

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