Q. \(x^2+2x-3=0\)

We want to solve the quadratic equation

\[x^2+2x-3=0.\]

Step 1: Identify the coefficients.

Compare \(x^2+2x-3=0\) to the standard form \(ax^2+bx+c=0\). Then:

\[a=1,\quad b=2,\quad c=-3.\]

Step 2: Factor the quadratic.

We look for two numbers whose product is \(ac\) and whose sum is \(b\).

\[ac=(1)(-3)=-3,\quad \text{and we want two numbers that sum to }2.\]

The numbers \(3\) and \(-1\) satisfy this because:

\[3+(-1)=2,\quad 3\cdot(-1)=-3.\]

Step 3: Write the factored form.

Using those numbers, rewrite the quadratic as:

\[(x+3)(x-1)=0.\]

Step 4: Use the zero-product property.

The zero-product property says: if a product equals zero, then at least one factor must be zero. So:

\[x+3=0 \quad \text{or} \quad x-1=0.\]

Step 5: Solve each simple equation.

First:

\[x+3=0 \Rightarrow x=-3.\]

Second:

\[x-1=0 \Rightarrow x=1.\]

Final answer:

\[x=-3 \quad \text{or} \quad x=1.\]