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Solving Quadratic Equations: Quadratic Formula

Learn how to solve quadratic equations using the **quadratic formula, x=−b±b2−4ac2ax=\frac{-b\pm\sqrt{b^2−4ac}}{2a}, and understand how many solutions there are. Let’s get started! 🚀

Solving Quadratic Equations: Quadratic Formula - introduction visual

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Quadratic formula for solving quadratic equations, and the discriminant for determining the number of real roots.Quadratic formula example with a = 3, b = 5, c = 2 solving 3x² − 5x + 2 = 0, giving roots x₁ = 1 and x₂ = 2/3.Solving a quadratic equation using the quadratic formula, with step-by-step breakdown and final solutions.

🛎️ The Quadratic Formula and the Discriminant

  • For any quadratic in the form ax2+bx+c=0ax^2+bx+c=0, the solutions are given by x=−b±b2−4ac2ax=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.
  • The part under the square root, b2−4acb^2-4ac, is called the discriminant and tells you how many solutions there are.
  • If the discriminant is negative, there are no real solutions, so you can stop calculating.

🛎️ Example: Using the Quadratic Formula

  • For 3x2−5x+2=03x^2-5x+2=0, substitute a=3a=3, b=−5b=-5, and c=2c=2 into the formula.
  • The discriminant is (−5)2−4×3×2=25−24=1(-5)^2 - 4 \times 3 \times 2 = 25 - 24 = 1, which means there are two different real solutions.
  • Substituting gives x=5±16x=\frac{5\pm1}{6}, so the solutions are x=1x=1 and x=23x=\frac{2}{3}.

🛎️ Example: Rearranging Before Using the Formula

  • First rearrange 6=−4x2+14x6=-4x^2+14x into the form ax2+bx+c=0ax^2+bx+c=0: 4x2−14x+6=04x^2-14x+6=0.
  • Simplify by dividing every term by 2 to get 2x2−7x+3=02x^2-7x+3=0, which makes the calculations much easier.
  • Substitute a=2a=2, b=−7b=-7, and c=3c=3 into the quadratic formula to find the solutions.

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