91Ó°ÊÓ

Quadratic equations often can be solved by factoring. Factoring is the process of breaking down an equation into simpler components called factors. These factors, when multiplied, give back the original equation.
For the quadratic equation \(x^{2}+4 x=3\), to start factoring, you first need to set the equation to zero. This means moving all terms to one side: \[ x^{2} + 4x - 3 = 0 \]
Next, you need to find two numbers that multiply to the constant term (-3) and add to the linear term (4). However, sometimes factoring directly may not yield correct solutions, as seen in the mistake in the steps.
That's where it becomes crucial to apply trial and error or use alternative methods like the quadratic formula to ensure correctness.

When factoring doesn't work or seems complicated, the quadratic formula is a reliable method to find solutions to any quadratic equation. The quadratic formula is given by: \[ x = \frac{{-b \ pm \ \sqrt{{b^2 - 4ac}}}}{2a} \]
For our equation \( x^2 + 4x - 3 = 0 \), the coefficients are:

• \( a = 1 \)
• \( b = 4 \)
• \( c = -3 \)

Substituting these into the quadratic formula:

• \[ x = \frac{{-4 \ pm \ \sqrt{{4^2 - 4(1)(-3)}}}}{2(1)} \]
• \[ x = \frac{{-4 \ pm \ \sqrt{{16 + 12}}}}{2} \]
• \[ x = \frac{{-4 \ \ pm \ \sqrt{28}}}{2} \]
• \[ x = \frac{{-4 \ pm \ 2\ \sqrt{7}}}{2} \]
• \[ x = -2 \ \ pm \ \sqrt{7} \]

This means the solutions are \( x = -2 + \sqrt{7} \) and \( x = -2 - \sqrt{7} \).

Always verify your solutions to ensure accuracy. Plug the solutions back into the original equation to see if they hold true.
For instance, let's check the solutions \( x = -2 + \sqrt{7} \) and \( x = -2 - \sqrt{7} \) in the original equation \( x^2 + 4x = 3 \).
Plugging \( x = -2 + \sqrt{7} \) into the left-hand side:

• \[ \left( -2 + \sqrt{7} \right)^2 + 4\left( -2 + \sqrt{7} \right) \]
• \[ = (-2 + \sqrt{7})^2 + 4(-2 + \sqrt{7}) \]
• \[ = 4 - 4\sqrt{7} + 7 - 8 + 4\sqrt{7} \]
• \[ = 3 \]

Similarly, checking \( x = -2 - \sqrt{7} \):

• \[ \left( -2 - \sqrt{7} \right)^2 + 4\left( -2 - \sqrt{7} \right) \]
• \[ = (-2 - \sqrt{7})^2 + 4(-2 - \sqrt{7}) \]
• \[ = 4 + 4\sqrt{7} + 7 - 8 - 4\sqrt{7} \]
• \[ = 3 \]

Both solutions satisfy the equation, ensuring they are correct.