Solving a Quadratic Equation Using the Quadratic Formula

The formula for solving a quadratic equation is the quadratic formula:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}x=2a−b±b2−4ac​​

Here, ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 is the standard form of a quadratic equation, where:

• $a$, $b$, and $c$ are coefficients.
• $x$ represents the unknown variable.
• The symbol $\pm$ indicates that there are generally two solutions: one using the plus sign and the other using the minus sign.

Example: Solve the quadratic equation 2×2+4x−6=02x^2 + 4x – 6 = 02x2+4x−6=0

1. Identify the coefficients:
   • $a=2$
   • $b=4$
   • $c=-6$
2. Substitute the coefficients into the quadratic formula:

x=−4±42−4(2)(−6)2(2)x = \frac{-4 \pm \sqrt{4^2 – 4(2)(-6)}}{2(2)}x=2(2)−4±42−4(2)(−6)​​

3. Simplify inside the square root:

x=−4±16+484x = \frac{-4 \pm \sqrt{16 + 48}}{4}x=4−4±16+48​​ x=−4±644x = \frac{-4 \pm \sqrt{64}}{4}x=4−4±64​​ x=−4±84x = \frac{-4 \pm 8}{4}x=4−4±8​

4. Calculate the two possible solutions:
   • First solution:

x=−4+84=44=1x = \frac{-4 + 8}{4} = \frac{4}{4} = 1x=4−4+8​=44​=1

• Second solution:

x=−4−84=−124=−3x = \frac{-4 – 8}{4} = \frac{-12}{4} = -3x=4−4−8​=4−12​=−3

Final Answer:

The solutions to the quadratic equation 2×2+4x−6=02x^2 + 4x – 6 = 02x2+4x−6=0 are $x=1$ and $x=-3$.